How Credit Constraints Impact Job Finding Rates,
Sorting & Aggregate Output∗
Kyle Herkenhoff
Gordon Phillips
Ethan Cohen-Cole
March 17, 2016
Abstract
We examine empirically and theoretically the effect of consumer credit limits on job
finding rates and the subsequent replacement earnings of displaced workers. Using new
administrative data, we find that in response to an increase in credit limits equal to
10% of prior annual earnings, medium-tenure displaced mortgagors take .15 to 3 weeks
longer to find a job and, conditional on finding a job, their earnings replacement rates
are 0 to 1.7% greater. They are also more likely to work at larger and more productive
firms. We construct a labor sorting model with credit to provide a structural estimate
of the duration and earnings replacement rate elasticities, which we find to be .2 weeks
and 1.8%, respectively. We use the model to assess what happens if consumer credit
limits contract during a recession. We find that when credit limits tighten during a
downturn, employment rises but both labor productivity and output exhibit weaker
recoveries. The tension between recovery speed and recovery health is due to the fact
that when limits tighten, low-asset, low-productivity job losers are unable to self-insure.
Subsequently, they search less thoroughly and take relatively more accessible jobs at
less productive firms. As a result, standard measures of sorting improve.
∗
Herkenhoff: University of Minnesota. Phillips: University of Southern California, University of Maryland, & NBER. Cohen-Cole: Econ One Research. We are grateful for comments from Mark Aguiar, Naoki
Aizawa, Sofia Bauducco, Lukasz Drozd, Fatih Guvenen, Jonathan Heathcote, Henry Hyatt, Miles Kimball,
Marianna Kudlyak, Jeremy Lise, Carlos Madeira, Ellen McGrattan, Lee Ohanian, Fausto Pena, Fabrizio
Perri, Victor Rios-Rull, Jean-Marc Robin, Thomas Sargent, Sam Schulhofer-Wohl, Rob Shimer, Jim Spletzer, Randy Wright, Ming Xu, and Moto Yogo as well as seminar participants at Bonn, Central Bank of Chile,
Census, CESifo, ES-Bocconi, FGV, Iowa, Michigan, Miami, Minneapolis Fed, NBER SI-Macro Perspectives,
Northwestern, Notre Dame, Philadelphia Fed, Richmond Fed, SAM, SED, SOLE/MEA, USC, and Virginia.
We thank Brian Littenberg and the Census for their hospitality and ongoing support. We thank Ming Xu for
excellent research assistance. Herkenhoff and Phillips thank the Washington Center for Equitable Growth for
generous funding. Cohen-Cole and Phillips thank the NSF (Grant No. 0965328) for funding and TransUnion
for providing credit data. The views expressed in this article are solely those of the authors.
1
Recent research by Kaplan and Violante [2014] has shown that many households, even
those with large amounts of wealth, have very little liquid assets. At the same time, many
of these households have significant amounts of credit access (Herkenhoff [2013]). This
generates a potentially important consumption smoothing role for consumer credit if ‘handto-mouth’ households lose their jobs. While much is known theoretically and empirically
about the impact of unemployment benefits on employment outcomes (inter alia Katz and
Meyer [1990], Ljungqvist and Sargent [1998], Acemoglu and Shimer [1998], Chetty [2008],
Mitman and Rabinovich [2012], and Hagedorn et al. [2013]), little is known about the role
consumer credit plays in the search decisions of unemployed households, and even less is
known about how this interaction affects the macroeconomy.1 How does consumer credit
affect job finding rates, replacement earnings, or the types of jobs workers take? How does
access to consumer credit affect the allocation of workers to firms, and what does this imply
for labor productivity, output, and employment dynamics?
By answering these questions, we make several contributions. Our empirical contribution is to merge 5 million individual credit reports with administrative employment records
to measure the impact of consumer credit access on job finding rates and re-employment
earnings of displaced workers. We find that being able to replace 10% more of prior annual
labor earnings with personal revolving credit allows medium-tenure displaced mortgagors
take .15 to 3 weeks longer to find a job, and, among those who find a job, they obtain a 0 to
1.7% greater annual earnings replacement rate. Moreover, households with greater access to
credit find jobs at larger firms and more productive firms. These results are consistent with
individuals using personal credit to fund longer unemployment spells so that they can search
and find better job matches; however, these results make an important point previously
overlooked in existing empirical studies which is that independent of realized borrowing, the
potential to borrow affects search decisions regardless if credit lines are actually drawn down.
To our knowledge, we are the first to measure the elasticity of non-employment durations
and the elasticity of earnings replacement rates with respect to consumer credit access.
Our theoretical contribution is to develop a general equilibrium labor sorting model
with consumer credit. Relative to existing labor sorting models such as Shimer and Smith
[2000], Shi [2001], Lise and Robin [2013] and Eeckhout and Sepahsalari [2014], our framework
1
The nascent but growing literature on the topic has focused on two mechanisms, the self-insurance role
of credit (e.g. Athreya and Simpson [2006], Herkenhoff [2013], Athreya et al. [2014]) and labor demand
effects of credit (e.g. Bethune et al. [2013], Donaldson et al. [2014], Bethune [2015]). The equally sparse
empirical literature on unemployment and borrowing is limited due to data constraints (Hurst and Stafford
[2004], Sullivan [2008], Herkenhoff and Ohanian [2012] among others) but recent inroads are being made
with new account data (Baker and Yannelis [2015], Gelman et al. [2015] among others).
2
generates interactions between heterogeneous credit histories and the allocation of workers
to firms. While our focus is on households, this framework is tractable enough that it can
be used by future researchers to study a variety of questions related to misallocation and
credit access, including credit access among firms.
In our model, heterogeneous credit-constrained workers accumulate human capital while
working. When unemployed, they direct their search, as in Menzio and Shi [2010, 2011], for
jobs among heterogeneous firms.2 Firms differ with respect to capital and produce output
by combining the human capital of workers with their own physical capital (for simplicity
we refer to firm capital as physical capital, but this may also be thought of as intellectual
capital). We assume supermodularity, meaning that firms with greater amounts of physical
capital produce more with workers who have greater amounts of human capital. We therefore
measure sorting in the model as the raw correlation coefficient between worker human capital
and firm physical capital.3 Which worker matches with which firm determines both output
and labor productivity in this economy, and therefore the ability of unemployed households
to self-insure, either through saving or borrowing, and search for higher physical capital jobs
has the potential to change the path of a recovery.
Using this new theoretic framework, we make two quantitative contributions. First, we
estimate the model using public data in order to provide a set of independent structural
estimates of the elasticity of durations and the elasticity of earnings replacement rates with
respect to consumer credit access. We find an elasticity of durations with respect to unused
credit of .15 (implying about a .2 week longer non-employment duration with a credit line
worth 10% of prior income) which falls at the low end of our reduced form estimates. We
also estimate an elasticity of earnings with respect to unused credit, among those who find
a job, of approximately 1.8%.
Second, given that the model produces earnings and duration elasticities that are consistent with the data, we use the model as a laboratory to answer the question of what
2
Related theoretical work includes sorting models with frictions (inter alia Teulings and Gautier [2004],
Bagger and Lentz [2008], Eeckhout and Kircher [2011], Hagedorn et al. [2012], Bonhomme et al. [2014]),
frictionless assignment models with borrowing constraints (Fernandez and Gali [1999], Legros and Newman
[2002], and Strauss [2013]), and occupational choice under credit constraints (inter alia Neumuller [2014],
and Dinlersoz et al. [2015]). Our work is also related to Lentz [2009], Krusell et al. [2010] and Nakajima
[2012a] who have studied the impact of savings on search decisions, and Guerrieri and Lorenzoni [2011]
among others have looked at the role household borrowing constraints play in models with frictionless labor
markets.
3
This measure is highly correlated with the Spearman Rank correlation coefficient. We also report the
Spearman Rank correlation coefficient for completeness.
3
happens to labor sorting, productivity, and the ensuing employment recovery if consumer
credit limits expand or contract during a recession. The main experiment we conduct using
the model is to tighten borrowing limits during the 2007-2009 recession and then study the
subsequent recovery. In particular, we simulate the 2007-2009 recession by feeding actual
total factor productivity residuals into the model. During the recession, we permanently
tighten borrowing limits, delivering roughly a 3 percentage point reduction in the fraction of
households borrowing, and a 1 percentage point reduction in the aggregate debt to income
ratio. Upon impact and throughout the recovery, the tighter credit limit depresses output
per worker (labor productivity) by .28 percentage points and decreases overall output by .11
percentage points.
We find that when debt limits tighten, however, standard measures of sorting improve
and remain elevated throughout the recovery. The reason is that constrained households,
who are also more likely to have low human capital, take jobs with low physical capital.
Households with savings, who are more likely to have high human capital, are able to continue to search thoroughly for jobs with higher physical capital. Since sorting measures the
correlation between human capital and physical capital, and since this correlation actually
increases when debt limits tighten, standard measures of sorting improve. Even though sorting improves, because the jobs that are being created during periods of tight debt limits are
less-physical-capital-intensive jobs, output falls. What disconnects the positive comovement
of sorting with labor productivity and output is the presence of firm investment and household credit constraints, which are typically absent in sorting models. These results, which to
our knowledge are new, raise important questions about the welfare implications of sorting
patterns.
Lastly, when debt limits tighten during the recession, employment recovers more quickly.
The mechanism is that when credit limits tighten, unemployed low-human-capital-borrowers
lose their ability to self-insure and take relatively abundant jobs at less-capital-intensive
firms. In other words, constrained households take lower quality jobs, relatively fast. As a
result, there is a tradeoff between the speed of the labor market recovery and the health of
the recovery, measured by labor productivity and output.
The paper proceeds as follows. We first describe our conceptual framework in Section
1. Section 2 describes the data. Sections 3 and 4 contain our empirical results. Section 5
presents the model, Section 6 describes the model estimation and structural estimates of the
duration elasticity, Section 7 conducts the main counterfactual exercise of tightening debt
4
limits, and Section 8 concludes.
1
Job Finding and Unsecured Credit
Unsecured credit allows unemployed households to augment today’s liquid asset position
by borrowing against future income. In McCall models of search, such as those studied by
Athreya and Simpson [2006] and Chetty [2008], access to liquid assets allows households to
search more thoroughly for higher wage jobs. While this mechanism is at the heart of the
unemployment insurance literature, there is limited evidence linking access to liquid assets
and job search decisions. In an influential paper, Chetty [2008] shows that workers who receive unemployment benefits take longer to find jobs, with the effect being strongest among
low wealth households. He also shows that unemployed households who receive severance
payments take significantly longer to find jobs. However, to our knowledge, there are no
existing studies documenting the way consumer credit limits impact unemployment durations, subsequent wage outcomes, or the characteristics of the firms where these households
ultimately take jobs. To fill this gap in the empirical literature, we test two hypotheses:
Hypothesis 1: Ceteris paribus, greater credit access among the unemployed increases
non-employment durations.
Hypothesis 2: Ceteris paribus, greater credit access among the unemployed increases
subsequent re-employment earnings.
It is important to note that because durations increase with greater credit access, the
theoretic prediction of credit access on earnings (including zeros) is ambiguous since those
who have more credit are taking longer to find a job and have zero earnings. However,
conditional on finding a job, we test whether those with greater credit access find higher
wage jobs (this is what we mean by re-employment earnings).
2
Data and Definitions
Our main data source is a randomly drawn panel of 5 million TransUnion credit reports
which are linked by a scrambled social security number to the Longitudinal Employment
and Household Dynamics (LEHD) database. All consumer credit information is taken from
5
TransUnion at an annual frequency from 2001 to 2008. The TransUnion data includes
information on the balance, limit, and status (delinquent, current, etc.) of different classes
of accounts held by individuals. The different types of accounts include unsecured credit as
well as secured credit on mortgages.
The LEHD database is a quarterly matched employer-employee dataset that covers 95%
of U.S. private sector jobs. The LEHD includes data on earnings, worker demographic
characteristics, firm size, firm age, and average wages. Our main sample of earnings records
includes individuals with credit reports between 2001 and 2008 from the 11 states for which
we have LEHD data: California, Illinois, Indiana, Maryland, Nevada, New Jersey, Oregon,
Rhode Island, Texas, Virginia, and Washington. Since job dismissal and reason of dismissal
are not recorded in the LEHD, we follow Jacobson et al. [1993] and focus on mass layoffs.4
We then define several labor market variables of interest. First, we define non-employment
duration to be the number of quarters it takes an individual to find a job following a mass
displacement.5 Non-employment duration therefore takes values ranging from 0 (indicating
immediate job finding) to 9 (all spells longer than 9 quarters of non-employment are assigned
a value of 9).6
Second, we define replacement earnings as the ratio of annual earnings 1 year after layoff
over annual pre-displacement earnings. Suppose a worker is displaced in year t, then we
define the replacement earnings ratio to be the ratio of annual earnings in the year after
layoff, in year t + 1, to the pre-displacement annual earnings, in year t − 1. To avoid
attributing the duration of non-employment with replacement earnings, when we measure
replacement earnings, we condition on households who have a full year of earnings in year
t + 1. We consider alternate definitions of earnings replacement rates that include zeros as
well as longer-term measures of replacement earnings (e.g. in year t + 2) in Appendix C.
In terms of TransUnion variables, we focus on revolving credit because it can be drawn
down on short notice following job loss and paid-off slowly over time without any additional loan-applications or income-checks. Our main measure of credit access is therefore
an individual’s unused credit limit across all types of revolving debt (excluding mortgage
4
Appendix A includes details on the identification of mass layoffs.
We follow Abowd et al. [2009] (Appendix A, Definitions of Fundamental LEHD Concepts) to construct
our measures of job accessions and employment at end-of-quarter.
6
Very few workers in our sample of displaced workers remain non-employed for longer than 4 quarters.
Changing the censored value to 8 or 10 has no impact on the results.
5
6
related revolving debt) over annual earnings, measured prior to displacement.7 We call this
ratio the ‘unused revolving credit limit ratio.’8 The main components of revolving credit
include bank revolving (bank credit cards), retail revolving (retail credit cards), and finance
revolving credit (other personal finance loans with a revolving feature). In Appendix B.2 we
use alternate measures of credit access prior to layoff including (i) credit scores, (ii) unused
revolving credit inclusive of HELOCs, and (iii) total secured and unsecured unused credit.
3
Empirical Approach
The goal of this section is to estimate the impact of credit access on employment outcomes of
displaced workers. While many authors, including Jacobson et al. [1993], have argued that
mass layoffs are exogenous to worker characteristics, credit access upon layoff certainly is
not. To solve this issue, we need to find a characteristic of workers that impacts credit limits
and only impacts employment prospects through its impact on credit limits. To isolate such
exogenous variation in credit limits, we use three sets of orthogonal instruments which we
discuss in detail in the following sections: (i) geographic instruments, (ii) bankruptcy flag
removals, and (iii) individual-specific account-age instruments.
3.0.1
Saiz Instrument
The first approach is to follow Saiz [2010] and Mian and Sufi [2012] who exploit variation in
geography to answer a wide variety of questions. The idea is to instrument the unused credit
limit ratio of workers with the geographic constraints of the MSA in which the worker lives
and then use this exogenous component of unused credit limits to measure the impact of
credit on various employment outcomes such as job finding rates and earnings replacement
rates. Mian and Sufi [2012] have laid much of the groundwork for this instrument by showing
that geographic constraints significantly impact house price growth as well as leverage and
are orthogonal to labor markets except through their impact on leverage (their samples
7
The reason that our main measure of credit access excludes mortgage related credit is because we want
to isolate the impact of credit access on employment, independent of housing wealth. Instead, to control
for the component of housing wealth that can be drawn down upon job loss, we include directly HELOC
limits and equity proxies as controls in our empirical analysis. Section 4.4 provides detailed discussion on
this point.
8
Appendix A includes details on the construction of this ratio.
7
always exclude construction workers and real-estate related sectors). Our analysis relies on
the arguments made in Mian and Sufi [2012], but, rather than focusing on realized leverage
(realized borrowing), the channel we emphasize is that geographic constraints impact house
price growth, and house price growth is a determinant of credit access, and in particular,
credit limits.
There are two reasons why house prices determine access to revolving credit: (i) workers
have more access to capital and are less likely to default, increasing the propensity of lenders
to extend any type of credit, and (ii) lenders expect workers to consume more, and therefore
offer more credit cards since they profit from transaction volume (not just balances). In the
first-stage regression, from a purely statistical point of view, we show that the Saiz [2010]
geographic constraint instrument is a strong predictor of the unused credit limit ratio of
individual workers for the 38 MSAs present in our sample. In the second stage regression,
the predicted unused credit limit ratios from the first stage are used to measure the impact
of credit on non-employment durations and annual earnings replacement rates.
Consider the sample of workers laid-off due to plant closure in year t. Let Di,t denote the
non-employment duration (in quarters and capped at 9 quarters) of individual i who is laid
off in year t. Let li,t−1 denote the unused limit ratio of individual i in year t − 1, the year
prior to layoff.9 Let Xi,t include a set of additional controls. The first-stage regression is to
predict the unused credit limit ratio as a function of the housing supply elasticity, si,t .
li,t−1 = πsi,t + BXi,t + ui,t
(1)
These first-stage estimates of π and B are used to isolate the exogenous component of
the unused credit limit ratio, ˆli,t−1 . The second stage regression is then used to estimate how
this exogenous variation in credit impacts employment outcomes such as duration, Di,t .
Di,t = γ ˆli,t−1 + βXi,t + i,t
(2)
We provide a detailed discussion of identification assumptions in Section 4.4, and in Appendix
B.1 use over-identification tests to show that the level component of unused credit which is
being isolated by the Saiz instrument satisfies the exclusion restriction.
9
Due to the frequency of the credit reports, we use annual credit limit information. Likewise, the earnings
information is annual earnings. Durations of non-employment are measured in quarters.
8
3.0.2
Bankruptcy Flag Removals
Our second strategy is to follow Musto [2004] who exploits the fact that bankruptcy flags
are removed after ten years, and this gives rise to a large exogenous increase in credit access
which does not reflect changes in underlying credit worthiness or unobserved quality of the
household.10 Consider the set of displaced workers who have a bankruptcy flag on their
record prior to displacement, and let Removali,t equal 1 if the worker has their bankruptcy
flag removed in the year of displacement. Our empirical approach is to compare those who
have their bankruptcy flag removed in the year of displacement (the treatment group) to
those whose flags remain on their record in the year of displacement (the control group). We
therefore implement the following specification in order to isolate the impact of increased
credit access on outcome variables such as duration, Di,t ,
Di,t = γRemovali,t + βXi,t + i,t
(3)
In this specification γ represents how much longer a displaced worker takes to find a job
if their flag is removed relative to the control group of households whose flag is not removed.
The key identifying assumption for this specification is that the treatment and control group
follow similar trends in non-employment durations and earnings replacement rates prior to
flag removal. Verifying the pretrends assumption for durations and earnings replacement
rates would require multiple mass displacements, which is very rare, but in a companion
paper, Herkenhoff et al. [2016], we show that bankrupt households whose bankruptcy flag is
not removed follow similar pretrends in terms of self-employment, earnings, size of employer,
etc. as bankrupt households whose flags are removed.
3.0.3
Gross and Souleles Instrument
The last instrument is based on the identification strategy of Gross and Souleles [2002] who
exploit the fact that credit card limits increase automatically as a function of the length of
time an account has been open.11 We exploit these time-contingent changes in credit access
10
Chapter 7 Bankruptcy flags (the most pervasive type of bankruptcy) remain on a worker’s credit report
for a statutory 10 years, whereas Chapter 13 flags are removed after 7 years. We cannot differentiate between
these types of flag removal, but what matters for us is that the removal is statutory.
11
As we discuss in Appendix B.1, the general mechanism is that credit issuers revise account limits based
on set time intervals. The subsequent limit revision is a function of credit scores, which themselves are an
9
by using the age of the oldest account as the instrument si,t for credit limits in the first
stage regression of equation (1). The main challenge to exogeneity for this isntrument is
that account ages are related to physical ages. Unlike credit scoring companies, however, we
observe physical age. We argue and provide supporting evidence with over-identification tests
that, after controlling for physical age as well as a host of other individual characteristics, that
account age satisfies the exclusion restriction. In Appendix B.1 we discuss this instrument
and the over-identification tests in more detail.
3.1
Samples
We use two samples in this paper.
i. Displaced Mortgagor Sample: Our first sample includes displaced workers with
mortgages who had at least 3 years of tenure at the time of displacement, worked
in a non-construction or non-real-estate industry, and worked at a firm with at least
25 employees. These are standard restrictions used in the literature, e.g. Davis and
Von Wachter [2011], to mitigate any issues associated with seasonal employment or weak
labor-force attachment. Given these criteria we end up with a sample (to the nearest
thousand) of 32,000 individuals.12 Given the way we identify displacements, and our
use of lagged credit prior to displacement, this sample covers the years 2002-2006.
ii. Displaced Bankrupt Sample: Our second sample includes all displaced households
who had a bankruptcy flag on their record in the year prior to displacement. We then
split this sample into a treatment group of 1,000 individuals whose flags are removed in
the year of displacement, and a control group of 17,000 individuals whose flags remain
on their records throughout the displacement. In order to garner enough observations
to disclose this sample, we had to drop the tenure requirement to 1 year. This sample
covers the years 2002-2006.
increasing function of account ages.
12
Census requires sample numbers to be rounded off to the nearest hundred to ensure no individual data
is disclosed or can be inferred. We round to the nearest thousand to allow for quicker disclosure of results.
10
3.2
Descriptive Statistics and Raw Correlations
Table 1 includes summary statistics for each sample considered in the paper. All variables
are deflated by the CPI, and the top 1% (and bottom 1% if the variable is not bounded
below) of continuous variables are winsorized. Column (1) of Table 1 shows that among all
displaced workers, the average age is 44.1 years old and they worked at their prior job for
about 5.9 years before the mass layoff. On average, their annual labor earnings were about
$56k prior to layoff. Workers can replace on average 45% of their prior annual labor earnings
with unused revolving credit.13 Workers involved in mass displacements take roughly 1.67
quarters to find a new job. Annual earnings replacement rates are 81% one year after mass
displacement, including zeros, similar to what Davis and Von Wachter [2011] find. Finally,
the age of the oldest account is approximately 15 years on average in our sample.
Column (2) of Table 1 shows that if we condition on households who have a job in the
year following displacement (i.e. in period t + 1), the average duration is .56 quarters, and
the average earnings replacement rate is 1.02 (meaning a full recovery of earnings). By
conditioning on employment, this sample drops earnings replacement rates equal to zero and
thus parses out any effects of duration on earnings.
Figure 1 plots the duration of non-employment by unused revolving credit to income
decile prior to layoff for the displaced mortgagor sample, and Figure 2 plots the earnings
replacement rate by unused revolving credit to income decile prior to layoff for those individuals in the displaced mortgagor sample who found a job in the year following displacement.
The deciles of unused revolving credit to income range from approximately zero to roughly
200%. In other words, those in the top decile can approximately replace 2x their annual income with revolving credit. Both figures reveal a generally monotone increasing relationship
between unused credit prior to layoff and both durations and earnings replacement rates,
with a pronounced rise in the last decile of unused credit.
Columns (3) and (4) of Table 1 summarize the displaced bankrupt sample. Our displaced
bankrupt sample is split into a treatment and control group. On average the demographic
characteristics between these two groups are quite close: imputed years of education is 13.8
years for those who have their flag removed (the treatment group) and 13.6 for those who do
not have their flag removed (the control group) and average tenure is 4 years in the treatment
13
The distribution of available credit is skewed. In the SCF, unused credit card limits to annual family
income among the unemployed peaks at 38% in 1998, and among the employed it peaks at 33% in 2007.
11
group and 3.9 in the control group. Those whose flags are removed are naturally older, with
an average age of 44 in the treatment group and an average age of 42.4 in the control group.
The treatment group earned $49k before displacement whereas the control group earned
$43.5k; while there are level differences in earnings, we show in a companion paper that in
general, those who have flag removals and those who do not have flag removals have parallel
earnings trends Herkenhoff et al. [2016]. The treatment group takes 1.73 quarters to find
a job on average after displacement, and their earnings replacement rate is 83%, whereas
the control group takes 1.53 quarters on average to find a job after displacement, and their
earnings replacement rate is 88%. We show in the regression analysis that follows that the
difference in durations is significant after adjusting for composition, whereas the difference
in earnings replacement rates is not.
4
Empirical Results
For each dependent variable, we show four sets of results corresponding to the baseline OLS
estimate, firstly, and then the three instruments, (i) geography, (ii) flag removals, and (iii)
account ages. In every specification, our vector of controls (Xi,t ) includes quadratics in age
and tenure as well as sex, race and education dummies, lagged annual income, cumulative
lagged earnings (to proxy for assets), 1-digit SIC industry dummies, lagged characteristics
of the previous employer including the size, age, and wage per worker, a dummy for the
presence of auto loans, an equity proxy (the highest mortgage balance observed less the
current balance), HELOC limits (to proxy for available housing wealth upon layoff), as well
as year dummies, the MSA unemployment rate, and MSA income per capita.
4.1
Duration Results
Table 2 illustrates the impact of unused credit on durations. Column (1) of Table 2 illustrates
the results from a simple OLS regression of duration on unused credit limits. The estimates
reveal a duration elasticity of .25, which implies that being able to replace 10% more of prior
annual income with unused credit is associated with an increase in duration of .3 weeks.14
In Column (2), we instrument the unused credit limit with the Saiz [2010] instrument. We
14
.3 weeks comes from multiplying the 10% increase in credit access to income by the OLS coefficient and
then multiplying again by 12 weeks per quarter (i.e. .3 weeks=.1*.25*12).
12
find a duration elasticity of 1.5 which implies that being able to replace 10% more of prior
annual income with unused credit allows workers to take roughly 1.8 weeks longer to find a
job. In Column (3) we instrument the unused credit limit with the Gross and Souleles [2002]
instrument. We find a duration elasticity of .6 which implies that being able to replace 10%
more of prior annual income with unused credit allows workers to take roughly .7 weeks
longer to find a job. Column (4) uses the displaced bankrupt sample, and shows the impact
of a derogatory flag drop on duration. Column (4) shows that if an individual’s bankruptcy
flag is dropped in the year of displacement, they take on average .178 quarters longer to find
a job (about 2 weeks longer), relative to the control group.
To convert the flag removal result into an elasticity, Table 3 shows that a bankruptcy
flag is associated with a contemporaneous increase in revolving credit limits equal to 7% of
prior annual income and an increase in credit scores of 140 pts. This implies an elasticity of
durations with respect to credit access of nearly 3 weeks (=.1*.178*12/.07). The size of this
point estimate suggests that there are non-linear effects of credit access on durations since
credit constrained households are reacting more per dollar of credit than less constrained
households. In Appendix C.2 we discuss non-linearities in our estimates in more detail.
Our estimates imply that $1 of additional unused credit limit is about half as potent for
unemployment durations as $1 of unemployment benefit. Being able to replace 5% of annual
earnings on a credit card is equivalent to a 10% increase in UI replacement rates for the
typical 6-month unemployment benefit. In the empirical UI literature, the impact of a 10%
increase in the UI replacement rate for 6 months is to increase unemployment durations by .3
to 2 weeks with the modal estimate lying between .5 and 1 for the US (see Nakajima [2012b]
and Card et al. [2015] for a summary of recent empirical and quantitative elasticities). Our
estimates imply an equivalent elasticity with respect to credit of .15 to 1.5 weeks.
For robustness, Appendix C.1 merges our sample with Schedule C tax records to adjust
the non-employment spells for self-employment. Appendix C.1 also uses the earnings gap
method to infer partial quarters of non-employment. Under either of these definitions of
non-employment duration, we find that the main results are robust.
13
4.2
Earnings Replacement Rates
Earnings Replacement Rates Including Zeros: Table 4 illustrates the impact of unused
credit on replacement rates of annual earnings, including zeros for those who dont find a job.
The point estimates in Column (1) through (4) of Table 4 show that the impact of additional
credit access on replacement earnings is statistically indistinguishable from zero. There are
two competing forces generating this result: (i) durations increase with more credit access,
depressing replacement earnings, (ii) of those who find a job, those who have more credit
access find higher wages, increasing replacement earnings. In the remainder of the section,
we isolate the second effect.
Earnings Replacement Rates Excluding Zeros: To avoid confounding annual replacement earnings with durations, in Table 5 we isolate the set of households who have positive
earnings in each quarter during the year after layoff. Table 5 reveals that conditional on
finding a job, those with greater credit access find higher wage jobs.
Table 5 shows that being able to replace 10% more of prior annual income increases the
earnings replacement rate of job finders by .3% in the OLS regressions, by 1.72% using the
Saiz Instrument, and by .8% using the Gross and Souleles instrument. Column (4) shows
that if an individual’s bankruptcy flag is dropped in the year of displacement, there is a
small negative, but statistically insignificant impact of credit access on earnings replacement
rates.
These results are, in general, in line with US estimates in the UI literature. Studies that
have considered the impact of unemployment benefits on re-employment earnings have found
positive and significant but mixed-magnitude effects in US data (see Addison and Blackburn
[2000] for a summary), whereas European studies have found both positive and insignificant
effects, as well as negative effects in one case (see Nekoei and Weber [2015] for a summary).
In Appendix C, we explore earnings replacement rates at longer horizons for this sample.
4.3
Size and Productivity of Firms
In this section, we show that among individuals who find a job in the year after displacement,
those with greater credit access are more likely to work at larger and more productive
firms. Our main dependent variables include an indicator function if the worker finds a
job at a firm in the 99th percentile of the firm size distribution or better (‘Large Firm
14
Dummy’), measured 1 year after displacement, as well as an indicator function if the worker
finds a job at a firm in the 75th decile of the wage-per-worker distribution (aggregate wage
bill divided by total employees), which is our proxy for productivity.15 These deciles were
chosen for comparability: i.e. firms in the 99th percentile of the size distribution comprise
approximately 1/3 of employment, and firms in the 75th percentile of the wage-per-worker
distribution comprise approximately 1/3 of employment. While not reported here, we find
similar effects using alternate cutoffs that are either narrower or broader.
Table 6 illustrates the impact of unused credit on the odds that a worker finds a job at
a firm in the 99th decile of the size distribution or greater. Being able to replace 10% more
of prior annual income increases the odds that a worker finds a job at a large firm by an
insignificant amount in both the OLS and Gross and Souleles regressions, by 4.7% using the
Saiz Instrument and by 3.9% following a bankruptcy flag removal.
Table 7 illustrates the impact of unused credit on the odds that a worker finds a job at a
firm in the 75th decile of the wage-per-worker (our proxy for labor productivity) distribution
or greater. Being able to replace 10% more of prior annual income increases the odds that
a worker finds a job at a productive firm by an insignificant amount in the OLS and Saiz
regressions, by 1.5% using the Gross and Souleles instrument, and 3% following a bankruptcy
flag removal.
While these results have mixed significance levels, the modal estimates are positive and
significant, implying that those with greater credit access prior to layoff are more likely to
be re-employed at larger and more productive firms.
4.4
Discussion of Identifying Assumptions for Saiz Instrument
There are two main challenges to exogeneity of the Saiz [2010] instrument: (i) aggregate
conditions, and (ii) housing wealth. We conduct a thought experiment to address the first
challenge. While Mian and Sufi [2012] argue that there is no correlation between the supply
elasticity and aggregate conditions except through leverage, if there is a correlation, it should
bias our results toward zero. Suppose MSAs with low supply elasticities have quickly rising
house prices and have better labor markets, then credit should expand and non-employment
durations should be shorter in those MSAs. This is the exact opposite of what our IV
15
What we call firms in the text are State Employment Identification Numbers (SEINs) in the LEHD.
SEINs aggregate all plants within a state.
15
estimates reveal.
To mitigate concerns about housing wealth, we include an equity proxy (the highest
mortgage balance ever observed less the current balance) and HELOC limits (home equity
lines of credit) in each table. When trying to capture wealth effects coming from house
price appreciation, HELOC limits isolate the relevant portion of housing wealth for job
loss episodes. HELOC limits indicate what portion of the home can be used as an ATM
immediately following a job loss. We argue that the HELOC credit limit just prior to a job
loss is a good proxy for access to liquid housing assets during the non-employment spell.
The idea is that whether the house is worth 200k or 220k should not affect short term job
search decisions directly. Only if the job loser can use the equity of the home to smooth
consumption, should the value of the home impact short term job search decisions.
Furthermore, since the average job loss spell in our data is quite short, it is unlikely that
a worker who is laid off will be able to secure additional home equity lines, or vacate the
home immediately and sell the house. As Piazzesi et al. [2015] show empirically, it takes
over 1 quarter for the median homeowner to sell their home, once it is listed. In our sample,
we do not see workers disproportionately selling their homes during or after layoff.
Moreover, in Appendix B.4, we conduct OLS regressions of unemployment duration on
unused credit, directly controlling for the OFHEO house price index. We show that the
inclusion of house prices does not affect our point estimates.
To further address the role of housing wealth and wealth more generally, we turn to the
Survey of Consumer Finances (SCF) in Appendix B.3. Using the 1998 to 2007 SCF surveys,
we show that the relationship in the SCF between non-employment duration and unused
credit card limits (the only limit available in the SCF) is similar to our IV estimates and
unaffected by the direct inclusion of self-report home values, liquid assets, or illiquid assets.
Lastly, we attempt to directly test the exogeneity condition of the Saiz and Gross and
Souleles instruments in Appendix B.1 by conducting over-identification tests. Section B.1
uses both the housing supply elasticity and the age of the oldest account to achieve overidentification. In all cases, the instruments pass over-identification tests at 5% significance
levels. These over-identification specifications also allow us to test an important assumption
which is that we estimate the impact of housing supply elasticities on levels on unused credit,
as opposed to changes. This approach isolates level components of credit access solely due to
geography, and the over-identification tests imply that the level component of credit being
16
isolated by the Saiz instrument satisfies the exclusion restriction.
4.5
Borrowing by Displaced Workers
One important point of our empirical section is that regardless of realized borrowing, the
potential to borrow affects job search decisions, regardless if the credit line is actually drawn
down. Workers know that if their buffer stock of liquid assets is depleted, they can borrow,
and this affects their job search decisions even if they never borrow. Existing work by Sullivan [2008] using the PSID and SIPP has shown that about 20% of workers borrow during
unemployment, and it is precisely low wealth workers who borrow during unemployment.
Unfortunately we do not have wealth information, but we plot the distribution of bankcard
borrowing among displaced workers. Figure 3, which is a smoothed density, plots the change
in bankcard balance among displaced workers 1 year after layoff relative to one year before
layoff. The graph reveals significant heterogeneity in borrowing responses among displaced
workers. Some workers borrow, consistent with Sullivan [2008], whereas some workers save,
also consistent with Sullivan [2008]’s regression results. As a result, the net amount borrowed among displaced workers is close to zero, consistent with recent findings (e.g. Gelman
et al. [2015]). However, this masks quite large and economically significant heterogeneity in
borrowing by displaced workers.
We further explore the role of borrowing by displaced workers in Figure 4 which illustrates
the change in real revolving debt in the year of layoff relative to 1 year before layoff as a
function of duration.16 Figure 4 shows that borrowing is a weakly increasing function of
unemployment duration, which suggests that those who were able to take the longest to find
a job were those who were able to borrow the most. The graph is a raw mean, and the
standard error is a standard error for the mean, so there is significant composition bias still
present in the graph. To address these concerns, Table 8 illustrates regression results for
the relationship between non-employment duration and borrowing, controlling for as many
characteristics of workers as possible. Column (1) omits controls, and Column (2) includes
controls. The coefficient in Column (2) implies that for every additional quarter of nonemployment, workers borrow on average $200, which is a relatively small average amount.
As discussed above, however, this regression masks the significant heterogeneity of how much
16
To obtain more power in the graph, durations are recoded to increase sample sizes, (i.e. durations of
length 0 or 1 are coded as 1, durations length 2 to 3 are coded as 3, 3-4=4,5-6=6,7-8=8, 9=9).
17
displaced workers borrow, which we are exploring in future research.
5
Model
To get around the inherently local nature of our IV estimates, we now turn to a structural
model which we use to obtain independent ‘global’ estimates of the unemployment duration
and earnings replacement rate elasticities. We then use the model to conduct our main experiment which is to consider how changes in aggregate borrowing limits impact the allocation
of workers to firms, output, and productivity.
Let t = 0, 1, 2, . . . denote time. Time is discrete and runs forever. There are three types of
agents in this economy. A unit measure of risk averse finitely-lived households, a continuum
of risk neutral entrepreneurs that run the endogenously chosen measure of operating firms,
and a unit measure of risk neutral lenders.
As in Menzio et al. [2012], there are T ≥ 2 overlapping generations of risk averse households that face both idiosyncratic and aggregate risk. Each household lives T periods deterministically and discounts the future at a constant rate β ∈ (0, 1). Every period households
first participate in an asset market where they make asset accumulation, borrowing, and
bankruptcy decisions. After the asset market closes, households enter the labor market
where they direct their search for jobs.17 Let ct,t+t0 and Lt,t+t0 respectively denote the consumption and hours worked of an agent born at date t in period t + t0 . The objective of a
household is to maximize the expected lifetime flow utility from non-durable consumption
and leisure.
"
Et
T
X
#
t0
β u(ct,t+t0 , 1 − Lt,t+t0 )
t0 =1
From this point on we will drop time subscripts and focus on a recursive representation
of the problem. We assume that labor is indivisible, such that the household consumes its
entire time endowment while employed L = 1, and vice verse for the unemployed.
Households are heterogeneous along several dimensions. Let b ∈ B ≡ [b, b] ⊂ R denote
17
The way directed search is modeled in this paper rules out the possibility that wage gains may simply
reflect differences in bargaining power and outside options.
18
the net asset position of the household, where b > 0 denotes that the household is saving, and
b < 0 indicates that the household is borrowing. Let h ∈ H ⊂ R+ denote the human capital
of the worker. Workers also differ with respect to the capital k ∈ K ⊂ R+ of the firm with
which they are matched, and with respect to their credit access status a ∈ {G, B} where
a = G denotes good standing, and a = B denotes bad standing. Let NT = {1, 2, . . . , T }
denote the set of ages.
The aggregate state of the economy includes three components: (i) total factor productivity (TFP) y ∈ Y ⊂ R+ and (ii) the borrowing limit b ⊂ R− , and (iii) the distribution of agents
across states µ : W, U × G, B ×B ×H ×K ×NT → [0, 1]. Let Ω = (y, b, µ) ∈ Y ×R− ×M
summarize the aggregate state of the economy where M is the set of distributions over the
state of the economy. Let µ0 = Φ(Ω, b0 , y 0 ) be the law of motion for the distribution, and
assume productivity and the borrowing limit follow a Markov process. It is important to
note that even though there is an exogenously imposed borrowing limit b, debt will be individually priced as in Chatterjee et al. [2007], and many workers will have ‘effective borrowing
limits’ where the bond price reaches zero well before b.
Let M (u, v) denote the matching function, and define the labor market tightness to be the
ratio of vacancies to unemployment. Since there is directed search, there will be a separate
labor market tightness for each submarket. In each submarket, there is a job finding rate
for households, p(·), that is a function of the labor market tightness θt (h, k; Ω), such that
t (h,k;Ω))
. On the other side of the market, the hiring rate for
p(θt (h, k; Ω)) = M (ut (h,k;Ω),v
ut (h,k;Ω)
firms pf (·) is also a function of the labor market tightness and is given by pf (θt (h, k; Ω)) =
M (ut (h,k;Ω),vt (h,k;Ω))
. Once matched with a firm, a worker produces f (y, h, k) : Y ×H×K → R+
vt (h,k;Ω)
and keeps a share α of this production.18
At the beginning of every period, households with debt positions b < 0 make a default
decision. In the present formulation, the default punishment is similar to Ch. 7 bankruptcy
in the United States. A household in bankruptcy has a value function scripted by B and
cannot save or borrow. With probability λ, a previously bankrupt agent regains credit access.
If a household is in good standing (i.e. they have regained credit access), its value function
18
This a similar assumption to Kaplan and Menzio [2013], and is only made for tractability purposes.
Directed search models with commitment to one submarket, including Shi [2001], find that firms optimally
post unique wages that are monotone in workers’ types, but other models in which firms do not commit to
any given submarket, such as Shimer [2001], find non-monotone wages in workers’ types within any given
job, in some cases. Empirically, wage profiles are concave in education and decreasing for higher levels of
education. We can allow for this by introducing flexible functional forms for production.
19
is scripted with a G, and the household can freely save and borrow.
The problem of an unemployed household in good standing is given below. To suppress
an additional state variable, we allow unemployment benefits z(k) to be a function of the
worker’s prior wage, but only through its dependence on k.19
h
UtG (b, h, k; Ω) = max
u(c,
1)
+
βE
max p(θt+1 (h0 , k̃; Ω0 ))Wt+1 (b0 , h0 , k̃; Ω0 )
0
b ≥b
k̃
i
+ 1 − p(θt+1 (h0 , k̃; Ω0 )) Ut+1 (b0 , h0 , k; Ω0 ) , t ≤ T
UTG+1 (b, h, k; Ω) = 0
Such that
c + qU,t (b0 , h, k; Ω)b0 ≤ z(k) + b
We assume that human capital abides by the following law of motion (note that the process
is indexed by employment status U ):
h0 = H(h, U )
And the shock processes and aggregate law of motion are taken as given:
y 0 ∼ F (y 0 | y),
b0 ∼ F (b0 | b),
µ0 = Φ(Ω, y 0 , b0 ),
Ω0 = (y 0 , b0 , µ0 )
(4)
For households who default, they are excluded from both saving and borrowing. There is an
exogenous probability λ that they regain access to asset markets:
h
UtB (b, h, k; Ω) = u(c, 1)+λβE max p(θt+1 (h0 , k̃; Ω0 ))Wt+1 (0, h0 , k̃; Ω0 )
k̃
i
+ 1 − p(θt+1 (h0 , k̃; Ω0 )) Ut+1 (0, h0 , k; Ω0 )
h
B
+ (1 − λ)βE max p(θt+1 (h0 , k̃; Ω0 ))Wt+1
(0, h0 , k̃; Ω0 )
k̃
i
B
0
0
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (0, h , k; Ω ) , t ≤ T
UTB+1 (b, h, k; Ω) = 0
19
Shocks to k during unemployment could proxy expiration of unemployment benefits.
20
Such that
c ≤ z(k)
and the law of motion for human capital and aggregates are taken as given. For households
in good standing, at the start of every period, they must make a default decision:
n
o
Ut (b, h, k; Ω) = max UtG (b, h, k; Ω), UtB (b, h, k; Ω) − χ
Let DU,t (b, h, k; Ω) denote the unemployed household’s default decision. Due to the finite
life cycle, a utility penalty of default, χ, is necessary to support credit in equilibrium.
A similar problem holds for the employed. The value functions are denoted with a W for
employed households, and at the end of every period, employed households face layoff risk
δ. If they are laid off, since the period we will ultimately use is 1 quarter, we must allow the
workers to search immediately for a new job.20 We relegate the employed value functions to
Appendix D.
5.1
Lenders
There is a continuum of potential lenders who are risk neutral and can obtain funds, without
constraint, at the risk free rate rf . Lenders may lend to households or firms. Recall e ∈
{E, U } denotes employment status. The price of debt for households must therefore satisfy
the inequality below:
qe,t (b0 , h, k; Ω) ≤
h
i
E 1 − De0 ,t (b0 , h0 , k 0 ; Ω0 )
1 + rf
(5)
Under free entry, the price of debt must yield exactly the risk free rate, rf , and this equation
holds with equality.
The price of debt for firms follows a similar form. For the sake of brevity, and the
necessity for additional notation, this bond price will be shown below in the firm section.
Since lenders earn zero profit for each contract in equilibrium, lenders are indifferent between
lending to a firm or a household.
20
This allows the model to match labor flows in the data.
21
5.2
Firms
There is a continuum of risk neutral entrepreneurs that operate constant returns to scale
production functions. The entrepreneurs invest in capital k ∈ K ⊂ R+ and post vacancies
to attract workers in the frictional labor market. We assume capital is denominated in units
of the final consumption good.
The entrepreneur, when attempting to create a firm, is subject to a financing constraint.
When a firm is not yet operational, the firm does not have access to perfect capital markets.
The firm must borrow the money, bf < 0 to finance the initial capital investment. We assume
the firm is not subject to the aggregate debt limit.21 If the firm fails to find an employee,
the firm defaults and the capital is lost.22
When deciding whether or not to post a vacancy, the firm solves the following problem.
It chooses capital k ∈ K and what types of workers, indexed by human capital and age
(h, t) ∈ H × NT , to hire. In the event that the worker is hired, the firm has access to perfect
capital markets and repays bf immediately. In the event that no worker can be found, the
firm defaults. Let Jt (h, k; Ω) be the profit stream of a firm that has k units of physical capital
and is matched with an age t worker with human capital h. Let qf,t (b, k, h; Ω) denote the
bond price faced by the firm. Then the problem a firm solves when attempting to recruit a
worker is given below (recall b is negative if borrowing),
κ ≤ max pf (θt (h, k; Ω))[Jt (h, k; Ω) + bf ] + (1 − pf (θt (h, k; Ω))) · 0
(6)
−k ≥ qf,t (bf , k, h; Ω)bf
(7)
k,h,t
such that
With free entry in the lending market, the price of debt must be given by (note that k is
21
This is because we want to isolate the impact of household credit limits on the macroeconomy. In future
work, we are exploring the role of firm constraints on sorting.
22
We are envisioning specific assets with low liquidation value, however, in Appendix G.4 we allow for an
explicit partial liquidation by the lender (capital is denoted in units of the final consumption good, and so
this amounts to capital reversibility).
22
implicitly related to bf in the equation above),
qf,t (bf , k, h; Ω) =
pf (θt (h, k; Ω))
1 + rf
(8)
Using the fact that Equation (6) holds with equality under free entry and that Equation (7)
must also hold with equality, the market tightness in each submarket which is entered with
positive probability is given by,
θt (h, k; Ω) = p−1
f
κ + (1 + rf )k
Jt (h, k; Ω)
!
(9)
For tractability, we assume that workers and firms split output according to a constant piecerate α. We assume the firm keeps a share 1 − α of its production, and workers receive the
remaining share α of production. Of that remaining output, firms must then pay a fixed
cost fc .23 The value function for the firm is given by,
Jt (h, k; Ω) = (1 − α)f (y, h, k) − fc + βE (1 − δ)Jt+1 (h0 , k; Ω0 ) ,
∀t ≤ T
JT +1 (h, k; Ω) = 0
There are three stark assumptions implicit in this value function, (i) zero liquidation value
of capital, (ii) static capital, and (iii) no on-the-job search. In Appendix G we allow capital
to have a nonzero liquidation value and we allow firms to dynamically invest in capital.
We do not explicitly model on-the-job search due to tractability (it would require firms
knowing workers’ asset policy functions, see Herkenhoff [2013] for a model with one sided
heterogeneity, credit, and OJS), but by allowing firms to invest in capital, we mitigate
workers’ incentives to switch jobs; in fact, with frictionless capital adjustment, firms set
capital to the surplus maximizing value and workers have no incentive to leave the firm.
5.3
Equilibrium: Definition, Existence and Uniqueness
Let x summarize the state vector of a household. An equilibrium in this economy is
a set of household policy functions for saving and borrowing ({b0e,t (x)}Tt=1 ), bankruptcy
23
The representative entrepreneur will make exactly zero profits across plants and over time, even if some
firms are temporarily making negative profits. When calibrating the model this fixed cost will serve to
generate a small surplus for firms, and help the model match quantitative features of the data.
23
({De,t (x)}Tt=1 ), and a capital search choice {kt (x)}Tt=1 , a debt price ({qe,t (x)}Tt=1 ) for both the
employed (e = W ) and unemployed (e = U ), a debt price for firms ({qf,t (x)}Tt=1 ), a market
tightness function θt (h, k; Ω), processes for aggregate shocks (y, b), and an aggregate law of
motion Φ(Ω, y 0 , b0 ) such that
i. Given the law of motion for aggregates, the bond price, and market tightness function,
households’ decision rules are optimal.
ii. Given the law of motion for aggregates and the bond price, the free entry condition in
the labor market (9) holds.
iii. Given household policy functions, the labor market tightness function, and the law of
motion for aggregates, the free entry conditions for lenders making loans to households
(5) and firms (8) both hold.
iv. The aggregate law of motion is consistent with household policy functions.
In what follows below, we use the same tools as Menzio and Shi [2011] to solve for a Block
Recursive Equilibrium in which policy functions and prices do not depend on the aggregate
distribution µ (even though it fluctuates over time and can be recovered by simulation).
However, policy functions still depend on aggregate productivity, y, and the borrowing limit,
b. As we show below, a Block Recursive Equilibrium exists in this economy, and thus to solve
the model economy, we only need to solve the first ‘block’ of the equilibrium i.-iii. ignoring
iv., and then we can simulate to recover the dynamics of µ. Furthermore, we establish that
certain classes of production functions yield uniqueness.
We begin with Proposition 5.1 which is the existence result for a Block Recursive Equilibrium. Without loss of generality, we set the firm fixed cost fc to zero.
Proposition 5.1. Assume that the utility function meets standard conditions (u0 > 0, u00 <
0, limc→0 u0 (c) = ∞, limc→∞ u0 (c) = 0, and u is invertible), the matching function is invertible
and constant returns to scale, and there is a bounded support (which can be non-binding) for
the choice set of debt b ∈ B ⊆ [b, b] and the capital of firms k ∈ K ⊆ [k, k], then a Block
Recursive Equilibrium exists.
Proof. Appendix E
24
A simple corollary follows in which one can establish the existence of an equilibrium with
debt.
Corollary 5.2. Under the hypotheses of Proposition 5.1, so long as χ > 0 and B contains
a neighborhood of debt around 0, a Block Recursive Equilibrium with credit exists.
Proof. Appendix E
Now, we turn to uniqueness. In Lemma (5.3) we provide sufficient conditions for the
economy to admit a unique, Block Recursive Equilibrium.
Lemma 5.3. In addition to the assumptions in Proposition 5.1, let the production function
be Cobb-Douglas, i.e. f (y, h, k) = yh1−a k a (0 < a < 1), let the matching function be given
1
1
by M (u, v) = u 2 v 2 , let χ → ∞ (no default for households), the value of leisure is zero,
and assume there is no uncertainty over human capital h, aggregate productivity y, or the
borrowing limit b. Then if the utility function is negative, increasing, and concave (e.g.
c1−σ −1
for σ > 1 or u(c) = −e−c ), the household labor search problem (equation (12)) admits
1−σ
a unique solution.
Proof. Appendix E
Lemma 5.3 demonstrates that for a broad range of production functions and utility
functions, the model admits a unique solution, and so there is no equilibrium selection
implicitly taking place in the computation below.24 Removing uncertainty in the proof is
only for the sake of closed form solutions to the firm problem, and as long as the utility
function of the household is additively separable in leisure, the proof holds.
6
Calibration
The parameters are calibrated so that the model’s stochastic steady state is consistent with
1970-2007 averages. Stochastic steady state means that aggregate total factor productivity
24
We use value function iteration over discrete grids to compute the model. Appendix H describes the
solution algorithm in detail.
25
(y) still fluctuates but that the borrowing limit (b) is constant forever.25 The period is set
to one quarter. We calibrate the productivity process to match the Fernald et al. [2012],
non-utilization adjusted total factor productivity series. The series is logged and band pass
filtered to obtain deviations from trend with periods between 6 and 32 quarters. Aggregate
productivity deviations are assumed to fluctuate over time according to an AR(1) process:
ln(y 0 ) = ρ ln(y) + 1 s.t. 1 ∼ N (0, σe2 )
Estimation yields ρ = 0.894 and σe = 0.00543, and the process is discretized using Rouwenhurst’s method.
We set the annualized risk free rate to 4%. In stochastic steady state, we set b = −.5,
which is non-binding for all agents in our simulations. We set the job destruction rate to a
constant 10% per quarter, δ = .1 (Shimer [2005]). For the labor market matching function,
we use a constant returns to scale matching function that yields well-defined job finding
probabilities:
u·v
∈ 0, 1)
M (u, v) = ζ
ζ
1/ζ
(u + v )
The matching elasticity parameter is chosen to be ζ = 1.6 as in Schaal [2012].
Preferences are given below, where η is the flow from leisure, and L=1 for employed
persons and L=0 otherwise:
u(c, 1 − L) =
c1−σ − 1
+ η(1 − L)
1−σ
We set the risk aversion parameter to a standard value, σ = 2. The life span is set to
T = 80 quarters (20 years), and newly born agents are born unemployed, with zero assets,
in good credit standing, and with a uniform draw over the grid of human capital. The
household share of income, α, is set to 23 , and the production function is Cobb-Douglas,
f (y, h, k) = yha k (1−a) with parameter a = 23 . The bankruptcy re-access parameter λ = .036
generates the statutory 7 year exclusion period.
The remaining 8 parameters including the discount factor β, the unemployment benefit
z, the utility penalty of bankruptcy χ, the entry cost of firms κ, the fixed cost of opera25
A long sequence of productivity shocks is drawn according to the AR(1) process for y and held fixed. A
large number of agents (N=30,000) is then simulated for a large number of periods (T=270 quarters, burning
the first 100 quarters). Averages are reported over the remaining 170 quarters across R = 10 repetitions.
26
tions fc , the flow from leisure η, the human capital appreciation p+∆ rate, and the human
capital depreciation p−∆ rate are calibrated jointly to match 8 moments: the fraction of
households with liquid asset to income ratios less than 1%, the immediate consumption loss
from unemployment, the bankruptcy rate, the unemployment rate, the relative volatility of
unemployment to productivity, the autocorrelation of unemployment, the wage growth of 25
year olds, and the long term consumption losses from layoff. We do not directly target the
duration elasticity or earnings replacement rate elasticity.
The household discount factor β = .988, which implies a discount rate of about 5% per
annum, is calibrated to match the fact that 25.4% of households have a ratio of liquid assets
to annual gross income less than one percent.26
The unemployment benefit is set to a constant, z(k) = .101 ∀k, in order to match
the observed consumption losses following job loss.27 This value of z yields an average UI
replacement rate of approximately 40% for the lowest human capital workers (Shimer [2005]),
but implies significantly lower UI replacement rates of 10% for higher human capital workers,
in line with Chodorow-Reich and Karabarbounis [2013].
The labor vacancy posting cost κ = .034 is chosen to target a mean U6 unemployment
rate of 8.9% which is the 1994-2007 average.28
We set the bankruptcy utility penalty χ = .077 to generate the average bankruptcy rate
in the US from 1970-2007 of approximately .1% per quarter.29
The processes for human capital are calibrated to generate 1.05% wage growth per quarter
while employed, as well as the long term consumption losses of displaced households.30 These
26
See Herkenhoff [2013]. The data is from the SCF (and it predecessor survey the Survey of Consumer
Credit). For each household, we sum cash, checking, money market funds, CDS, corporate bonds, government
saving bonds, stocks, and mutual funds less credit card debt over annual gross income. We take the mean
of this liquid asset to income ratio across households in each survey year, and then we average over 1970 to
2007 to arrive at the moment.
27
Browning and Crossley [2001] find 16% consumption losses after 6 months of unemployment for Canadians, and as they explain, scaling food consumption losses in Gruber [1994] results in 15% consumption
losses in the year of layoff for US households in the PSID. We therefore target a 15% consumption loss
from the quarter prior to initial displacement until the end of the 1st year of layoff, 4 quarters after initial
displacement.
28
Since there is no concept of “marginally-attached” workers or part-time employment in the model, U6
is a better measure of unemployment for the model. The data is available from 1994:Q1 to present.
29
The bankruptcy rate is .41% per annum from 1970-2007 according to the American Bankruptcy Institute
(accessed via the Decennial Statistics).
30
Our measure of wage growth in the data is the median 2-year real-income growth rate for households
aged between 25 and 30 in the PSID between 2005 and 2007. In the data, the median growth rate among
27
processes are governed by two parameters p−∆ and p+∆ .

h − ∆ w/ pr. p
−∆ if unemployed
H(h, U ) = h0 =
h
w/ pr. 1 − p−∆ if unemployed

h + ∆ w/ pr. p
+∆ if employed
H(h, W ) = h0 =
h
w/ pr. 1 − p+∆ if employed
In the calibration below, the grid for human capital, h ∈ [.5, .6, .7, .8, .9, 1], as well as the
step size, ∆ = .1, between grid points are taken as given. Our estimates are p−∆ = .143
and p+∆ = .077, which produce similar human capital processes as Ljungqvist and Sargent
[1998]. Once every year-and-a-half, unemployed agents in the model expect to fall one rung
on the human capital ladder. This implies between 10% to 20% earnings losses (depending
on the initial human capital), which is smaller than the 30% per year Ljungqvist and Sargent
[1998] target.
In terms of the flow utility of leisure, we follow most of the quantitative search and
matching literature by setting η to target a labor market moment. We choose η = .237
to match the autocorrelation of unemployment since the flow utility of leisure determines
unemployed households’ willingness to remain out of work.
We calibrate the fixed cost of operations for firms fc = .100, which determines how
sensitive firms are to productivity shocks, to match the observed volatility of unemployment
to productivity.
Table 9 summarizes the parameters, and Table 10 summarizes the model’s fit relative to
the targeted moments. As we show in the next section, the model will succeed at replicating
the new empirical facts on debt and duration of unemployment; however, the model is
unable to match several moments, largely due to the inclusion of business cycle moments
as targets in the calibration. The fundamental tension in the model is between generating
this subset of households was 8.8% (we condition on at least 1k of earnings in each year). Converting this
estimate to quarters yields a quarterly income growth rate of 1.05%. Assuming agents are born at 25, our
model measure of wage growth is at the midpoint of that interval, measured among 27.5 year olds in the
model. Using the 2005-2011 PSID, we calculate a full consumption recovery (1% higher relative to pre-layoff
consumption) 2 years after layoff for unemployed households who have zero duration spells. For distressed
layoffs, Saporta-Eksten [2013]’s estimates long-run consumption losses of approximately 8% two years after
initial displacement. We take the average of these two estimates and target a 3% consumption loss.
28
borrowing/bankruptcies and matching the business cycle facts: intuition would suggest that
lowering the discount factor would be the best way to generate borrowing/bankruptcies
but doing so only exacerbates the models ability to match business cycle facts. The more
impatient are agents, the more they want to work immediately, regardless of productivity,
which dampens business cycle dynamics. Because of the two sided heterogeneity, we cannot
deploy the simple fixes in Hagedorn and Manovskii [2008]; raising the value of leisure will, at
best, make only the lowest-human-capital agents indifferent between working and not, and
reducing firm surplus will, at best, make only the lowest-capital firms sensitive to productivity
movements. The remainder of the distribution of workers and firms will not, in general,
respond to productivity shocks. We discuss this more in Appendix F.
6.1
Non-Targeted Moments: Model Estimates of Duration and
Earnings Replacement Rate Elasticities
In this section, we use the model generated policy functions to estimate the duration and
replacement earnings elasticities with respect to credit access.31 Since the debt pricing
schedule does not have an explicit credit limit, we define the credit limit to be the maximum
of either the level of debt where the bond interest rate first exceeds 30% (denote this level of
debt b30 (·)) or the exogenous debt limit b.32 Therefore, we define the credit limit for an agent
with state vector x as L(x) = min{−b30 (x), −b}. We isolate newly laid off agents (let Iδ
denote this set of agents, and let Nδ denote its cardinality), and then we compute each agent’s
optimal search decision under loose (b = bL ) and tight exogenous debt limits (b = bH > bL ),
ceteris paribus. What makes this calculation feasible is that the policy function of each
agent is contingent on the realization of Ω which includes the exogenous debt limit b. So
at each decision node, encoded in this policy function is the search decision of the agent
if debt limits tighten as well as if debt limits remain slack. What makes this experimental
design valid is the block recursive nature of the model; the menu of job choices faced by
the household is not a function of b. This allows us to determine the impact of changing
debt limits, holding all else constant, including the set of jobs from which households can
31
We calculate the duration and replacement elasticities using 30,000 agents simulated for 200 periods
(burning the first 100 periods), while holding the aggregate state fixed at y = 1, and defining bH =-.1 and
bL =-.5. Agents hold the same rational beliefs over the transition rate Pb between bH and bL as Section 7.
32
Since virtually no agents borrow at rates above 30%, this is a robust definition of the credit limit.
Changing the threshold to the point at which agents face a 50% interest rate does not alter the results
either.
29
choose.33 We compute the change in unemployment duration, weighted by the distribution
of job losers after moving from an exogenous limit b = bL to b = bH as follows,34
∆Durt =
X Dur(bi,t , hi,t , ki,t ; yt , b ) − Dur(bi,t , hi,t , ki,t ; yt , b )
H
L
N
δ
i∈I
δ
t +bt )
as the change in the unused credit to income ratio that the agent faces if
Define ∆(L
Yt−1
the exogenous debt limit is tightened.35 The model implied duration elasticity is therefore
given by,
∆Durt
= 0.15
dur = ∆(Lt +bt )
Yt−1
In other words, if unused credit to income increased by 10%, then agents would take .2 weeks
longer to find a job. This falls in the lower range of our IV estimates. However, the elasticity
calculated in the model is a ‘global’ elasticity and is conceptually different from the local
average treatment effect identified by the IV.
Next, we calculate the elasticity of replacement earnings with respect to credit, including
zeros. Let ei ∈ {W, U } denote employment status and I be the indicator function. Then deP
I(e =W )∗4αf (yt ,hi,t ,k∗ (bi,t ,hi,t ,ki,t ;yt ,b))+0∗(I(ei =U ))
fine Rt (b) as the earnings replacement rate, Rt (b) = N1δ i∈Iδ i
4αf (yi,t−1 ,hi,t−1 ,ki,t−1 )
The model implied replacement earnings elasticity is therefore given by,
Rep =
Rt (bH ) − Rt (bL )
= −.024
∆(L+b)
Yt−1
The model produces a near-zero, slightly negative, earnings replacement rate elasticity of
-.026, whereas in the data, the earnings replacement rate elasticity is positive and lies anywhere between zero and +.22. To understand why this is the case, we can decompose
earnings losses into two offsetting components: (i) access to additional credit depresses job
finding rates which tends to lower replacement earnings, and (ii) access to additional credit
33
The intuition is simple and is formally shown in the existence proof. JT (h, k; y) = f (y, h, k) does not
depend on b, and working back, neither does Jt (h, k; y) for arbitrary t. Therefore, using the free entry
condition, θt (h, k; y), which pins down the menu of operating submarkets, will not either.
34
The expected duration is based on the 1-quarter ahead implied job finding rate, based on the search
policy function. In quarters, for large M, the expected duration is given by, Dur(bt , ht , kt ; yt , bH ) =
PM
∗
∗
(m−1)
.
m=1 mp(θt (ht , k (bt , ht , kt ; yt , bH ); y, bH )) ∗ (1 − p(θt (ht , k (bt , ht , kt ; yt , bH ); yt , bH )))
∆(Lt +bt )
35
Let
Yt−1
denote
earnings
prior
to
layoff.
Define
=
Yt−1
P
(L(hi,t ,ki,t ;yt ,bH )+bi,t ∗I(bi,t <0))−(L(hi,t ,ki,t ;yt ,bL )+bi,t ∗I(bi,t <0)))
1
Nδ
i∈Iδ
4αf (yi,t−1 ,hi,t−1 ,ki,t−1 )
30
increases the capital intensity of submarkets searched by agents which tends to raise replacement earnings. We can compute each of these components separately. Define the job finding
P
rate for agents as JFt (b) = N1δ i∈Iδ p(θt (hi,t , k ∗ (bi,t , hi,t , ki,t ; yt , b); yt , b)). Then the model
implied job finding elasticity is given by,
JF =
JFt (bH ) − JFt (bL )
= −0.11
∆(L+b)
Yt−1
This implies that when debt limits expand by 10% of prior annual income, job finding rates
fall by 1.1% as workers can better self-insure while searching more thoroughly for jobs. This
tends to decrease the replacement earnings of agents, since unemployed workers have an
earnings replacement rate of zero.
Turning to the second component of replacement earnings, define the capital intensity
P
rate of submarkets in which agents search as Kt (b) = N1δ i∈Iδ k ∗ (bi,t , hi,t , ki,t ; yt , b)dµ. Then
the model implied capital intensity elasticity is given by,
K =
Kt (bH ) − Kt (bL )
= .27
∆(L+b)
Yt−1
In other words, being able to replace 10% more of prior income with credit allows agents in
the model to search in submarkets with 2.7% greater intellectual or physical capital intensity.
This tends to increase the replacement earnings of agents. The combination of the two effects,
namely the negative influence of job finding rates and positive influence of capital intensity
on replacement earnings, yields the near-zero replacement earnings elasticity observed in the
model.
Next, we calculate the elasticity of replacement earnings with respect to credit, among
job finders. Let Ie (b) denote the set of job finders at the end of period t. Let Nδ,e denote
the cardinality of Iδ ∩ Ie (b), which is the set of laid off households who find a job at the
end of period t. Define replacement earnings among this set of households as Rt,e (b) =
P
4αf (yt ,hi,t ,k∗ (bi,t ,hi,t ,ki,t ;yt ,b))
+bt,e )
1
. Lastly, define ∆(LYt,e
to be the change in credit
i∈I
∩I
(b
)
Nδ,e
4αf (yi,t−1 ,hi,t−1 ,ki,t−1 )
e L
δ
t−1,e
limits to income of those who find a job at the end of period t under borrowing limit bL .36
The model implied replacement earnings elasticity, among the employed, is therefore given
36 ∆(Lt,e +bt,e )
Yt−1,e
=
1
Nδ,e
P
(L(hi,t ,ki,t ;yt ,bH )+bi,t ∗I(bi,t <0))−(L(hi,t ,ki,t ;yt ,bL )+bi,t ∗I(bi,t <0)))
.
4αf (yi,t−1 ,hi,t−1 ,ki,t−1 )
∆(Lt +bt )
denominator, and are very similar using Yt−1 .
i∈Iδ ∩Ie (bL )
are insensitive to our choice of
31
The results
by,
Rep,e =
Rt,e (bH ) − Rt,e (bL )
= .18
∆(Lt,e +bt,e )
Yt−1,e
This implies that in the model, among job finders, being able to replace 10% more of prior
income with credit results in a 1.8% greater earnings replacement rate, which falls toward
the high end of our IV estimates. In summary, the model’s estimates of the duration and
earnings replacement rate elasticities are in line with our IV estimates.
7
Main Quantitative Experiment
Based on the model’s success at replicating key non-targeted micro moments, we now aggregate across individual agents to explore how credit access impacts the macroeconomy. In
particular, we study the way changes in borrowing limits impact the path of output, labor
market sorting and therefore productivity during the 2007-2009 recession. We do so by comparing aggregate outcomes across two economies, both of which have the same beliefs about
debt limit transitions Pb :
1. Tight Debt Limit Economy: The debt limit tightens from b = −.5 (a non-binding
value) to b = −.1 in 2008-Q4 (the first quarter in which the aggregate consumer credit
limit declined37 ), and stays there permanently. As we document in Table 11, the new
debt limit b = .1 delivers roughly the same magnitude decline in debt to income (DTI)
as the data.
2. Constant Debt Limit Economy: The debt limit b = −.5 remains constant throughout
the simulation.
In the experiments below, both economies are simulated in their ergodic stochastic steady
states with the non-binding debt limit, b = −.5, for a large number of periods. We then
feed in a realized set of shocks that replicates, as approximated on a grid, the path of the
Fernald et al. [2012] productivity residuals from 1974-Q1 to 2012-Q4. The borrowing limit
is held constant at b = −.5 through 2008-Q4 in both economies, for simplicity. In 2008-Q4,
one economy has the limit tighten to b = −.1, and it remains there permanently.
37
2014 Q1 New York Fed Consumer Credit Report, Page 7, time series entitled ‘Credit Limit and Balance
for Credit Cards and HE Revolving.’
32
We impose that both economies have the same beliefs over debt transitions. Let pl,l
be the probability of remaining in the ‘low’ debt limit state, b = −.1, and let ph,h be the
probability of remaining in the ‘high’ debt limit state, b =
! −.5. Then the transition matrix
pl,l
1 − pl,l
for the debt limit b is given by Pb =
.
1 − ph,h
ph,h
Agents understand that if the debt limit tightens, it is permanent, so we set pl,l = 1.
And, agents also understand that once every 34 years (from 1974 to 2008), debt limits will
tighten, so we set ph,h = .9926. On average, therefore, agents are rational.
7.1
Model Results
Figure 5 illustrates the path for the exogenous component of productivity y and the path
for the borrowing limit b. These are the two inputs in the experiment. Each plot contains
two dashed lines that correspond to differing degrees of debt limit tightening. The dashed
blue line corresponds to the economy where limits tighten to b = −.1, and, for the sake of
robustness, we also include the dash-dot red line which corresponds to the economy where
limits tighten to b = −.2 (this delivers a drop in DTIs half as severe as the b = .1 economy).
Firstly, Table 11 illustrates what the tightening of debt limits does to borrowing in
the model economies. The model economies see reductions in the fraction of households
borrowing of 3.09 percentage points, and 1.21 percentage points, respectively. Economy-wide
debt to income ratios fall by 1.09 percentage points and .53 percentage points respectively.
In the data, the fraction of households that stopped borrowing fell by 6.77 percentage points
from 2007 to 2010 (measured in the SCF) while the debt to income ratio fell by .86 percentage
points from 2007 to 2010 (measured in the SCF).38
Figure 6 plots the percentage change in employment during the 2007-2009 recession across
the economy with a tighter debt limit versus the economy with a fixed debt limit. When
debt limits tighten, employment tends to increase, persistently. The mechanism is that with
looser credit limits, unemployed households borrow to smooth consumption while thoroughly
searching for capital-intensive jobs. If debt limits tighten, they lose their ability to self-insure,
and, as a result, take low-capital-intensity jobs that are relatively quick to find. In other
words, when limits tighten, low-asset job losers take relatively less productive employment
38
While not reported here for the sake of space, the bankruptcy rate reaches .97% in the model in the
quarter in which limits are tightened, which is in line with ABI bankruptcies per capita.
33
opportunities. This introduces a strong tension between recovery speed and recovery health,
as workers find jobs more quickly but these jobs are of lower quality.
As Figure 9 shows, the aggregate capital stock held by entrepreneurs, which is a function
of existing matches and new matches, drops severely relative to the economy in which debt
limits are held constant. This is entirely driven by new entrepreneurial entrants posting
more vacancies in submarkets with less capital, and constrained households searching for
jobs in those submarkets. The time it takes for the aggregate capital stock to recover to
its pre-recession levels is as much as 6 quarters longer in the economy in which debt limits
tighten.
Because households become more constrained and take jobs in which there is less capital per unit of labor, Figure 7 shows that measured labor productivity, defined as output
over employment, declines when debt limits are tightened. The economy in which debt
limits tighten the most has .28 percentage points lower labor productivity as compared to
the economy with constant debt limits, and this productivity gap persists throughout the
recovery. In Appendix G, when we allow for capital investment to change over the course
of a match, the labor productivity decline is slightly less pronounced. The relative labor
productivity decline reaches .24 percentage points 16 quarters after the start of the recession
as firms subsequently invest more in capital in existing matches during the recovery.
While the impact of tighter debt limits on capital per worker is unambiguous, the impact
of tighter debt limits on aggregate output is theoretically ambiguous: households find jobs
faster, but the jobs workers find are less productive. However, Figure 8 shows that quantitatively the reduction in capital per worker is so severe that output falls by .11 percentage
points.
The mechanism at the heart of the output decline involves a reallocation of workers from
high capital firms to low capital firms. To understand this reallocation in greater detail,
we now turn to standard measures of sorting. Figure 10 plots the percentage change in the
correlation between human capital, h, and firm capital, k, during the 2007-2009 recession.
Figure 11 plots the corresponding percentage change in the Spearman rank correlation coefficient between human capital, h, and firm capital, k, as well (workers are ranked by h,
and firms are ranked by k, and the Spearman Rank correlation coefficient is the resulting
correlation between the numeric ranks of workers and firms). The raw correlation coefficient
between worker human capital and firm physical capital is approximately +.33.
34
Figures 10 and 11 show that in the economy in which debt limits are tighter, these
standard measures of sorting improve. The mechanism behind this sorting improvement is
that in the economy in which debt limits tighten, unemployed agents with low-human-capital
cannot borrow to smooth consumption while thoroughly searching for jobs. Therefore, they
take jobs that are less-capital-intensive, but more abundant. On average, since low human
capital workers are less productive (recall the assumption of supermodularity), tighter debt
limits force these ‘low quality’ workers to take ‘low quality’ jobs. As such, standard measures
of sorting improve, even as output falls, since they do not take into account the investment
decisions of firms. In this economy, these standard measures of sorting are not good proxies
for either productivity or output, even with a supermodular production function.
The fact that firms can invest or that workers can save are standard assumptions in most
neoclassical business cycle models but are often ignored in search theoretic models. The fact
that sorting moves in the opposite direction of output and productivity under these mild
assumptions raises important questions about the welfare implications of sorting patterns
derived from search theoretic models with linear utility or under the assumption of fixed
firm types.
7.2
Evidence on Countercyclical Sorting
The empirical literature on the cyclicality of sorting is very thin. In a model with linear
utility, Lise and Robin [2013] find similar countercyclical sorting patterns among certain
subgroups of households. Likewise, Bagger et al. [2013] compute long-run sorting trends,
and their time series reveal that sorting accelerated during the Danish recession in the early
1990s, and then declined in the tranquil 2000s. On the other hand, Moscarini and Vella
[2008] use the CPS to study occupational sorting, and they find that there is less occupational
sorting in recessions. In general, the available data supports the broad patterns observed in
the model and over the business cycle, but much research remains to be done on the topic.
7.3
Robustness: Capital Investment and Liquidation Value
We conduct two robustness exercises in Appendix G. First, we allow for the entrepreneurs
to invest in capital over time, mitigating concerns about both quits and on-the-job-search.
With costless adjustments to entrepreneur capital, there would never be a reason to quit or
35
change jobs. We find that our main results are largely unchanged, but the ability to invest in
capital during recoveries marginally dampens the response of capital, productivity, output,
and sorting to business cycle shocks. Second, we allow for a liquidation value of firm capital,
and again, the main predictions of the model still hold.
8
Conclusions
In this paper, we provide the first estimates of the impact of credit constraints on job finding
rates and subsequent replacement wages of displaced workers. Using new administrative
data, we find that medium-tenure displaced mortgagors, in response to being able to replace
10% of their annual income with revolving credit, take .15 to 3 weeks longer to find a job
but obtain an earnings replacement rate that is 0 to 1.7% greater.
We then develop, to our knowledge, the first labor sorting model with credit. We estimate
the model on a set of independent public moments in order to derive structural estimates
of the duration and replacement rate elasticities, yielding estimates of approximately .2 and
1.8%, respectively. We then use the model as a laboratory to understand how fluctuations
in debt limits impact productivity, output, and employment. We find that tighter debt
limits during recessions may increase employment during the recovery, but depress both
productivity and output. This tension between the speed of recovery and health of recovery
is at the heart of the mechanism: tighter debt limits force constrained households to cut
their job search short, taking relatively unproductive jobs that are more abundant.
The estimates and quantitative exercises in this paper have implications for the way
both policy-makers and economists think about the optimal provision of unemployment
insurance (Acemoglu and Shimer [1998], Shimer and Werning [2005], Chetty [2008], Mitman
and Rabinovich [2011], Hsu et al. [2014]) and the response of labor markets to monetary
policy (Gornemann et al. [2012], Auclert [2014]). The fact that increases in credit access can
actually reduce job finding rates by easing household credit constraints brings into question
the ability of the Federal Reserve Bank to effectively meet the dual mandate of “maximum
employment, stable prices and moderate long-term interest rates.” Furthermore, the theory
we advance in this paper is tractable enough that it can be used by future researchers to
study a variety of questions related to misallocation and credit access, including credit access
among firms.
36
We view this paper as the beginning of a research agenda which uses new micro data
and new theory to understand how consumer credit impacts the allocation of households to
firms. The next step in our research agenda is to measure the impact of consumer credit
constraints on the other side of the market, on the Schedule C entrepreneurs and the workers
they hire.
References
John M Abowd, Bryce E Stephens, Lars Vilhuber, Fredrik Andersson, Kevin L McKinney, Marc
Roemer, and Simon Woodcock. The lehd infrastructure files and the creation of the quarterly
workforce indicators. In Producer Dynamics: New Evidence from Micro Data, pages 149–230.
University of Chicago Press, 2009.
Daron Acemoglu and Robert Shimer. Efficient unemployment insurance. Technical report, National
Bureau of Economic Research, 1998.
John T Addison and McKinley L Blackburn. The effects of unemployment insurance on postunemployment earnings. Labour Economics, 7(1):21–53, 2000.
Kartik Athreya, Juan M Sánchez, Xuan S Tam, and Eric R Young. Labor market upheaval, default
regulations, and consumer debt. Federal Reserve Bank of St. Louis Working Paper Series,
(2014-002), 2014.
K.B. Athreya and N.B. Simpson. Unsecured debt with public insurance: From bad to worse.
Journal of Monetary economics, 53(4):797–825, 2006.
Adrien Auclert. Monetary policy and the redistribution channel. Technical report, MIT mimeo,
2014.
Jesper Bagger and Rasmus Lentz. An equilibrium model of wage dispersion with sorting. In 2008
Meeting Papers, number 271. Society for Economic Dynamics, 2008.
Jesper Bagger, Kenneth L Sørensen, and Rune Vejlin. Wage sorting trends. Economics Letters,
118(1):63–67, 2013.
Scott R Baker and Constantine Yannelis. Income changes and consumption: Evidence from the
2013 federal government shutdown. Available at SSRN 2575461, 2015.
Zach Bethune. Consumer credit, unemployment, and aggregate labor market dynamics. Technical
report, Working Paper, 2015.
Zach Bethune, Guillaume Rocheteau, and Peter Rupert. Aggregate unemployment and household
unsecured debt. 2013.
Stéphane Bonhomme, Thibaut Lamadon, and Elena Manresa. A distributional framework for
matched employer employee data. 2014.
Martin Browning and Thomas F Crossley. Unemployment insurance benefit levels and consumption
changes. Journal of Public Economics, 80(1):1–23, 2001.
Charles Calomiris, Stanley D Longhofer, and William Miles. The (mythical?) housing wealth effect.
Technical report, National Bureau of Economic Research, 2009.
David Card, Andrew Johnston, Pauline Leung, Alexandre Mas, and Zhuan Pei. The effect of
unemployment benefits on the duration of unemployment insurance receipt: New evidence from
37
a regression kink design in missouri, 2003-2013. Technical report, National Bureau of Economic
Research, 2015.
S. Chatterjee, D. Corbae, M. Nakajima, and J.V. Rı́os-Rull. A quantitative theory of unsecured
consumer credit with risk of default. Econometrica, 75(6):1525–1589, 2007.
R. Chetty. Moral hazard vs. liquidity and optimal unemployment insurance. Technical report,
National Bureau of Economic Research, 2008.
Gabriel Chodorow-Reich and Loukas Karabarbounis. The cyclicality of the opportunity cost of
employment. Technical report, National Bureau of Economic Research, 2013.
Lawrence J Christiano, Martin S Eichenbaum, and Mathias Trabandt. Unemployment and business
cycles. Technical report, National Bureau of Economic Research, 2013.
Steven Davis and Till Von Wachter. Recessions and the costs of job loss. 2011.
Emin Dinlersoz, Henry R Hyatt, and Hubert P Janicki. Who works for whom? worker sorting in
a model of entrepreneurship with heterogeneous labor markets. US Census Bureau Center for
Economic Studies Paper No. CES-WP-15-08, 2015.
Jason Roderick Donaldson, Giorgia Piacentino, and Anjan Thakor. Bank capital, bank credit, and
unemployment. 2014.
Jan Eeckhout and Philipp Kircher. Identifying sortingin theory. The Review of Economic Studies,
page rdq034, 2011.
Jan Eeckhout and Alireza Sepahsalari. Unemployment risk and the distribution of assets. Technical
report, Manuscript, 2014.
John Fernald et al. A quarterly, utilization-adjusted series on total factor productivity. Federal
reserve bank of San Francisco working paper, 19:2012, 2012.
Raquel Fernandez and Jordi Gali. To each according to? markets, tournaments, and the matching
problem with borrowing constraints. The Review of Economic Studies, 66(4):799–824, 1999.
Michael Gelman, Shachar Kariv, Matthew D Shapiro, Dan Silverman, and Steven Tadelis. How
individuals smooth spending: Evidence from the 2013 government shutdown using account data.
Technical report, National Bureau of Economic Research, 2015.
Nils Gornemann, Keith Kuester, and Makoto Nakajima. Monetary policy with heterogeneous
agents. 2012.
David B Gross and Nicholas S Souleles. Do liquidity constraints and interest rates matter for
consumer behavior? evidence from credit card data*. The Quarterly journal of economics, 117
(1):149–185, 2002.
Jonathan Gruber. The consumption smoothing benefits of unemployment insurance. Technical
report, National Bureau of Economic Research, 1994.
Veronica Guerrieri and Guido Lorenzoni. Credit crises, precautionary savings, and the liquidity
trap. Technical report, National Bureau of Economic Research, 2011.
M. Hagedorn and I. Manovskii. The cyclical behavior of equilibrium unemployment and vacancies
revisited. 2008.
Marcus Hagedorn, Tzuo Hann Law, and Iourii Manovskii. Identifying equilibrium models of labor
market sorting. Technical report, National Bureau of Economic Research, 2012.
Marcus Hagedorn, Fatih Karahan, Iourii Manovskii, and Kurt Mitman. Unemployment benefits
and unemployment in the great recession: the role of macro effects. Economics working paper,
University of Pennsylvania, 2013.
Song Han and Geng Li. Household borrowing after personal bankruptcy. Journal of Money, Credit
and Banking, 43(2-3):491–517, 2011.
38
K.F. Herkenhoff. The impact of consumer credit access on unemployment. Manuscript, 2013.
K.F. Herkenhoff and L.E. Ohanian. Foreclosure delay and us unemployment. Federal Reserve Bank
of St. Louis Working Paper 2012-017A, 2012.
Kyle F Herkenhoff, Gordon Phillips, and Ethan Cohen-Cole. The impact of consumer credit access
on employment, earnings and entrepreneurship. Manuscript, 2016.
Joanne W Hsu, David A Matsa, and Brian T Melzer. Positive externalities of social insurance:
Unemployment insurance and consumer credit. Technical report, National Bureau of Economic
Research, 2014.
E. Hurst and F. Stafford. Home is where the equity is: mortgage refinancing and household
consumption. Journal of Money, Credit and Banking, pages 985–1014, 2004.
Louis S Jacobson, Robert J LaLonde, and Daniel G Sullivan. Earnings losses of displaced workers.
The American Economic Review, pages 685–709, 1993.
Greg Kaplan and Guido Menzio. Shopping externalities and self-fulfilling unemployment fluctuations. Technical report, National Bureau of Economic Research, 2013.
Greg Kaplan and Giovanni L Violante. A model of the consumption response to fiscal stimulus
payments. Econometrica, 82(4):1199–1239, 2014.
F. Karahan and S. Rhee. Housing and the labor market: The role of migration on aggregate
unemployment. University of Pennsylvania, mimeo, 2011.
Lawrence F Katz and Bruce D Meyer. Unemployment insurance, recall expectations, and unemployment outcomes. The Quarterly Journal of Economics, 105(4):973–1002, 1990.
Per Krusell, Toshihiko Mukoyama, and Ayşegül Şahin. Labour-market matching with precautionary
savings and aggregate fluctuations. The Review of Economic Studies, 77(4):1477–1507, 2010.
Patrick Legros and Andrew F Newman. Monotone matching in perfect and imperfect worlds. The
Review of Economic Studies, 69(4):925–942, 2002.
Rasmus Lentz. Optimal unemployment insurance in an estimated job search model with savings.
Review of Economic Dynamics, 12(1):37–57, 2009.
Jeremy Lise and Jean-Marc Robin. The macro-dynamics of sorting between workers and firms.
2013.
L. Ljungqvist and T.J. Sargent. The european unemployment dilemma. Journal of Political Economy, 106(3):514–550, 1998.
G. Menzio and S. Shi. Block recursive equilibria for stochastic models of search on the job. Journal
of Economic Theory, 145(4):1453–1494, 2010.
G. Menzio and S. Shi. Efficient search on the job and the business cycle. Journal of Political
Economy, 119(3):468–510, 2011.
Guido Menzio, Irina A Telyukova, and Ludo Visschers. Directed search over the life cycle. Technical
report, National Bureau of Economic Research, 2012.
Atif R Mian and Amir Sufi. What explains high unemployment? the aggregate demand channel.
Technical report, National Bureau of Economic Research, 2012.
Kurt Mitman and Stanislav Rabinovich. Pro-cyclical unemployment benefits? optimal policy in an
equilibrium business cycle model. 2011.
Kurt Mitman and Stanislav Rabinovich. Do changes in unemployment insurance explain the emergence of jobless recoveries? 2012.
Giuseppe Moscarini and Francis G Vella. Occupational mobility and the business cycle. Technical
report, National Bureau of Economic Research, 2008.
39
D.K. Musto. What happens when information leaves a market? evidence from postbankruptcy
consumers*. The Journal of Business, 77(4):725–748, 2004.
Makoto Nakajima. Business cycles in the equilibrium model of labor market search and selfinsurance*. International Economic Review, 53(2):399–432, 2012a.
Makoto Nakajima. A quantitative analysis of unemployment benefit extensions. Journal of Monetary Economics, 59(7):686–702, 2012b.
Arash Nekoei and Andrea Weber. Does extending unemployment benefits improve job quality?
2015.
Seth Neumuller. Inter-industry wage differentials revisited: Wage volatility and the option value of
mobility. Available at SSRN 2143814, 2014.
Monika Piazzesi, Martin Schneider, and Johannes Stroebel. Segmented housing search. Technical
report, National Bureau of Economic Research, 2015.
Albert Saiz. The geographic determinants of housing supply. The Quarterly Journal of Economics,
125(3):1253–1296, 2010.
Itay Saporta-Eksten. Job loss, consumption and unemployment insurance. 2013.
Edouard Schaal. Uncertainty, productivity and unemployment in the great recession. Federal
Reserve Bank of Minneapolis, mimeo, 2012.
Shouyong Shi. Frictional assignment. i. efficiency. Journal of Economic Theory, 98(2):232–260,
2001.
R. Shimer. The cyclical behavior of equilibrium unemployment and vacancies. American economic
review, pages 25–49, 2005.
Robert Shimer. The assignment of workers to jobs in an economy with coordination frictions.
Technical report, National Bureau of Economic Research, 2001.
Robert Shimer and Lones Smith. Assortative matching and search. Econometrica, 68(2):343–369,
2000.
Robert Shimer and Iván Werning. Liquidity and insurance for the unemployed. Technical report,
National Bureau of Economic Research, 2005.
David Strauss. Financial development and sorting reversals-a theory of structural change. 2013.
J.X. Sullivan. Borrowing during unemployment. Journal of Human Resources, 43(2):383–412, 2008.
Coen N Teulings and Pieter A Gautier. The right man for the job. The Review of Economic
Studies, 71(2):553–580, 2004.
40
Table 1: Summary Statistics for Displaced Mortgagor Sample and Displaced Bankrupt Sample (Source: LEHD/TransUnion)
(1)
(2)
Mortgagor Sample
All
Employed
at t+1
Age
44.1
43.4
Tenure
5.9
5.9
Imputed Years of Education
13.9
13.9
Lagged Annual Earnings
$56,108
$57,733
Lagged Unused Revolving Credit to Income
0.45
0.41
Lagged Unused Total Credit to Income
0.77
0.71
Duration of Non-Employment (In Quarters)
1.67
0.56
Replacement Rate (Annual Earnings Year t+1/Annual 0.81
1.02
Earnings Year t-1)
Lagged Months Since Oldest Account Opened
181.7
179.0
Observations (Rounded to 000s)
32000
21000
(3)
(4)
Bankrupt Sample
Treatment Control
44.0
4.0
13.8
$48,893
0.12
0.30
1.73
0.83
42.4
3.9
13.6
$43,527
0.07
0.22
1.53
0.88
165.8
1000
163.0
17000
Notes. Sample selection criteria in Section 3.1.
Table 2: Dependent Variable is Duration. Column (1) OLS, Column (2) IV using the Saiz
instrument, Column (3) IV using the Gross and Souleles Instrument, Column (4) OLS with
Bankruptcy Flag Drop. (Source: 2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
———————–Dependent Variable is Duration———————–
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
0.249***
1.513**
0.514***
(0.0258)
(0.630)
(0.115)
Flag Drop (d)
0.178**
(0.0792)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.048
0.101
0.134
0.033
Angirst Pischke FStat Pval
0.000138
0
Round N
32000
32000
32000
18000
Notes. Std. errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Col. (1), (3), and (4) use robust std.
errors, and Col. (2) uses clustered std. errors at MSA Level. Unused Revolving Credit to Income measured
1 year prior to layoff. Demographic controls include quadratic in age & tenure, race, sex and education
dummies as well as year & auto loan dummies. Industry controls include 1-digit SIC dummies and lagged
size, age, and wage per worker of prior firm. MSA controls include real per capita GDP and the MSA
unemployment rate. Lagged earnings controls include both lagged real annual earnings, and cumulative
lagged real annual earnings to proxy for assets. Equity proxy is highest observed mortgage balance less
current mortgage balance. HELOC limits include combined home equity limits.
41
Table 3: Impact of Bankruptcy Flag Removal on Credit Access, OLS. Column (1) Dependent
Variable is Revolving Credit Limit to Income, and Column (2) Dependent Variable is Credit
Score. (Source: 2002-2006 LEHD/TransUnion)
(1)
Credit Limit to Inc
Flag Drop (d)
0.0689***
(0.00962)
Demographic, Industry, MSA, & Y
Lagged Earnings Controls
HELOC Limits and Equity Proxy
Y
R2
0.056
Round N
18000
(2)
Credit Score
140.1***
(6.257)
Y
Y
0.126
18000
Notes. Robust Std. errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Same control definitions as
Table 2.
Table 4: Dependent Variable is Replacement Rate, Measured 1 Year After Layoff Relative
to 1 Year Before Layoff. (Source: 2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
———————–Dependent Variable is Replacement Rate———————–
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
-0.00531
-0.129
0.0169
(0.00437)
(0.148)
(0.0176)
Flag Drop (d)
-0.0230
(0.0140)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.044
0.101
0.134
0.047
Angirst Pischke FStat Pval
0.000138
0
Round N
32000
32000
32000
18000
Notes. Same as Table 2.
42
Table 5: Dependent Variable is Replacement Rate, Measured 1 Year After Layoff Relative
to 1 Year Before Layoff. Sample Includes Those With Jobs 1 Year After Layoff. (Source:
2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
——— DV is Replacement Rate, Among Employed at t+1 ———
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
0.0332***
0.172**
0.0847***
(0.00344)
(0.0872)
(0.0139)
Flag Drop (d)
-0.0126
(0.0111)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.082
0.0872
0.117
0.146
Angirst Pischke FStat Pval
2.48e-05
0
Round N
21000
21000
21000
12000
Notes. Same as Table 2.
Table 6: Dependent Variable is Dummy if Firm is in 99th Decile of Size Distribution or
Greater (‘Large Firm Dummy’). Sample Includes Those With Jobs 1 Year After Layoff.
(Source: 2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
—————–Dependent Variable is Large Firm Dummy—————–
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
-0.000220
0.471**
0.0282
(0.00505)
(0.234)
(0.0257)
Flag Drop (d)
0.0392**
(0.0176)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.061
0.0872
0.117
0.056
Angirst Pischke FStat Pval
2.48e-05
0
Round N
21000
21000
21000
12000
Notes. Same as Table 2.
43
Table 7: Dependent Variable is Dummy if Firm is in 75th Decile of Wage Per Worker
Distribution or Greater (‘Productive Firm Dummy’). Sample Includes Those With Jobs 1
Year After Layoff. (Source: 2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
—————–Dependent Variable is Productive Firm Dummy—————–
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
-0.00375
-0.251
0.153***
(0.00459)
(0.322)
(0.0248)
Flag Drop (d)
0.0295*
(0.0164)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.229
0.0872
0.117
0.217
Angirst Pischke FStat Pval
2.48e-05
0
Round N
21000
21000
21000
12000
Notes. Same as Table 2.
Table 8: Dependent Variable is Change in Real Revolving Debt 1 Year After Layoff Minus
1 Year Before Layoff. (Source: LEHD/TransUnion)
(1)
Change in Revolving Debt
Duration of Unemployment
Constant
Demographic, Industry, MSA,
Lagged Earnings Controls
HELOC Limits and Equity Proxy
R-squared
Round N
48.93
(54.67)
2,676***
(153.5)
& N
(2)
Change in Revolving Debt
196.9***
(55.37)
-1,857
(2,236)
Y
N
3.17e-05
32000
Y
0.025
32000
Notes. Robust Std. errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Same control definitions as
Table 2.
44
Table 9: Summary of Model Parameters.
Variable
Value
ρ
0.894
σe
rf
δ
ζ
σ
α
a
λ
b
0.00543
4%
10%
1.6
2
0.66
0.66
0.036
-0.5
Variable
Value
κ
z
p−∆
p+∆
β
fc
η
χ
0.034
0.101
0.143
0.077
0.988
0.100
0.237
0.077
Pre-Calibrated
Description
Autocorrelation of Productivity Process
Std. Dev. Of Productivity Process
Annualize Risk Free Rate
Quarterly Layoff Rate
Matching Function Elasticity
Risk Aversion
Household share of income
Cobb-Douglas Labor Share
Bankruptcy Re-Access
Non-binding debt limit
Jointly-Estimated
Description
Firm Entry Cost
UI
Depreciation Rate of Human Cap.
Appreciation Rate of Human Cap.
Discount Factor
Fixed Cost
Flow Utility of Leisure
Bankruptcy Utility Penalty
Table 10: Model Calibration
Model
Target
Variable
Value
Source
Unemployment Rate
8.93%
Consumption Drop 1 Yr Af- 0.846
ter Layoff
Consumption Drop 2 Yrs 0.971
After Layoff
8.90%
0.84
κ
z
0.034
0.101
.97
p−∆
0.143
Quarterly Income Growth
Rate 25yo
Fraction of Households with
Liquid assets to Income
Ratio<1%
Vol U/ Vol y
Autocorr Unempl
Bankruptcy rate
1.078%
1.05%
p+∆
0.077
BLS, U6 1994-2007
Browning & Crossley
(2001)
Saporta-Eksten
(2013) / PSID 20052011
PSID, 2005-2007
0.093
0.254
β
0.988
SCF, 1974-2007
2.686
0.730
0.01%
9.5
0.94
0.10%
fc
η
χ
0.100
0.237
0.077
Shimer (2005)
Shimer (2005)
ABI, 1970-2007
45
Table 11: Reduction in Borrowing When Borrowing Limit Tightens, Model v. Data.
∆ Fraction of
HHs Borrowing
∆ DTI
Debt limit tightened from b = −.5 to b = −.1
-3.09%
-1.09%
Debt limit tightened from b = −.5 to b = −.2
-1.21%
-0.53%
Data
-6.77%
-0.86%
Notes: All Differences Computed using 2007 and 2010 SCF. DTI is Change Unsecured Revolving Consumer Credit to Annual Family Income. Fraction Borrowing
Change is Difference in Fraction of Households Carry Positive Balances. Means
Weighted Using Survey Weights. Model statistics calculated as difference in average
of quarterly values over same corresponding years.
46
47
Figure 3: Change in Real Revolving Debt between Figure 4: Change in Real Revolving Balance beYear of Layoff Minus 1 Year Before Layoff (Source: tween Year of Layoff Minus 1 Year Before Layoff
LEHD/TransUnion)
as Function of Non-Employment Duration (Source:
LEHD/TransUnion)
Figure 1: Non-Employment Duration by Unused Revolv- Figure 2: Replacement Earnings 1 Year After Layoff (Ining Credit to Income Decile, prior to layoff (Source: cluding 0s) by Unused Revolving Credit to Income Decile,
LEHD/TransUnion)
prior to layoff (Source: LEHD/TransUnion)
48
Figure 7: Labor Productivity, 2007-2009 Recession
Figure 5: Experiment Input: Exogenous Aggregate Productivity (y) and Borrowing Limit b, 2007-2009 Recession
Figure 8: Aggregate Output
Figure 6: Percentage Change in Employment Per Capita
49
Figure 11: Spearman Rank Correlation Coefficient Between Human Capital (h) and Firm Capital (k)
Figure 9: Aggregate Firm Capital, 2007-2009 Recession
Figure 10: Correlation Between Human Capital (h) and
Firm Capital (k)
Online Appendix Not For Publication
50
A
Data Appendix
Employer reports are based on the ES-202 which is collected as part of the Covered Employment and Wages (CEW) program (run by BLS). One report per establishment per quarter
is filed. On this form, wages subject to statutory payroll taxes are reported.
The employment records are associated with a firm’s State Employment Identification
Number (SEIN). This is an identifier based on an employer within a given state, and it is,
in general, not an identifier of the establishment of the worker. Minnesota is the only state
to collect establishment identifiers, and in all other states, an imputation based on place-ofwork is used to generate establishment level identifiers. In general, workers are included in
the dataset if they earn at least one dollar from any employer.
The Quarterly Census of Employment and Wages (QCEW) contains firm level data
which is collected in each state. This dataset includes information on industry, ownership,
and worksite.
The demographic data in the LEHD comes from the 2000 census as well as social security
records, and tax returns. These are linked by social security number with the unemployment
insurance data. In the LEHD, social security numbers are not present, rather there is a
scrambled version called a Protected Identification Key (PIK).
The main demographic information database is the Person Characteristic File (PCF).
Information on sex, date of birth, place of birth, citizenship, and race are included here.
A.1
Identifying Mass Layoffs
To identify mass layoffs, we combine data from the Longitudinal Business Dynamics (LBD)
database on establishment exits with the LEHD. In each state, employers are assigned a
State Employment Identification Number (SEIN) in the LEHD database. This is our unit
of analysis for mass layoffs. We define a mass layoff to occur when an SEIN with at least 25
employees reduces its employment by 30% or more within a quarter and continues operations,
or exits in the LEHD with a contemporaneous plant exit in the LBD. In California, we do not
have LBD establishment exit information, however. To ensure that the there was actually a
mass layoff, we then verify that fewer than 80% of laid-off workers move to any other single
SEIN using the Successor Predecessor File (SPF). This allows us to remove mergers, firm
51
name-changes, and spin-offs from our sample.
A.2
TransUnion Variables
The unsued revolving credit limit ratio is defined as,
(Total Revolving Credit Limit - HELOC credit limit) - (Total Revolving Balance - HELOC balance)
Lagged Annual Earnings
‘Total Revolving Credit Limit’ corresponds to the TransUnion variable ‘Revolving High
Credit/Credit Limit.’ ‘Revolving High Credit/Credit Limit’ is constructed as the sum of
the ‘High Credit/Credit Limit’ across all types of revolving debt. The ‘High Credit/Credit
Limit’ is defined as the actual credit limit if such a limit is recorded or the highest historical
balance if no credit limit is recorded. ‘HELOC credit limit’ is the sum across all available
HELOC credit limits, and ‘HELOC balance’ is the sum across all available HELOC balances.
B
B.1
Robustness Checks
Over-Identification Tests
In this section we discuss the assumptions underlying the Gross and Souleles instrument, and
we use the fact that we have two different instruments in order to conduct over-identification
tests. In summary, the Saiz and Gross and Souleles instruments pass identification tests at
the 5% significance level, suggesting that they are satisfying exclusion restrictions. However,
there is no true test of exogeneity.
For the Gross and Souleles instrument to be valid, it must be both relevant and exogenous. What makes the Gross and Souleles instrument relevant is that a large component,
approximately 15%, of a credit score is solely based on length of credit history.39 By simply
having an account open, you credit score increases and affects your credit limits.40 Empir39
See ‘Your Credit Score,’ prepared by Fair Isaac Corporation and available from http://www.
consumerfed.org/pdfs/yourcreditscore.pdf
40
Limits are then revised upward as credit scores increase. As Gross and Souleles [2002] explain, credit
issuers revise account limits regularly, and the length of time an account was open is a determinant of these
credit limit revisions. They write, “many issuers will not consider (or are less likely to consider) an account
for a line change if it has been less than six months or less than one year since the last line change” (p.7).
52
ically, the first stage is very strong, as evidenced by the small p-values in the Stock-Yogo
weak identification tests.
For the age of the oldest account to be a valid instrument for credit limits, it must not
only be a strong determinant of credit limits, but it can only have an impact on employment
prospects through credit limits (exogeneity). The main challenge to exogeneity is that the
age of an account is related to the physical age of the individual. Since the age of an account
is how scoring companies proxy for physical age, by conditioning on physical age (which we
observe but scoring companies do not), we are able to isolate changes in credit scores simply
due to variation in account age, that have nothing to do with physical age.41
Lastly, since we have multiple instruments, we show that both the Gross and Soueles
instrument and Saiz instrument pass over-identification tests. For individual i, li,t is unused
credit, gi,t is the age of the oldest account, si,t is the supply elasticity, and Xi,t is a vector
of characteristics including quadratics in age and tenure. To conduct the over-identification
tests, we implement the following specifications:
li,t−1 = π1 si,t + π2 gi,t + BXi,t + ui,t
(10)
These first-stage estimates of π1 , π2 and B are used to isolate the exogenous component
of the unused credit limit ratio, ˆli,t−1 . The second stage regression is then used to estimate
how this exogenous variation in credit impacts employment outcomes such as duration, Di,t .
Di,t = γ ˆli,t−1 + βXi,t + i,t
(11)
The main idea behind the over-identification tests is to predict ˆli,t using one instrument,
e.g. si,t , and then verify that the other instrument, gi,t , is uncorrelated with the resulting
residuals.
Table 12 illustrates the main results using both the Saiz and Gross and Souleles instruments in the first stage. We see that the instruments pass the over-identification tests (in
particular, we implement Hansen’s J-test). The null is that the instruments are valid, and so
larger p-values indicate the fact that we cannot reject the null that the instruments are valid.
41
The reason they do this is that the Equal Credit Opportunity Act bans the use of age, race, or sex in
determining credit score, and so the credit scoring companies use account age as a proxy for physical age.
53
The instruments pass the J-test at both 1% and 5% significance levels. In terms of point estimates, Table 12 shows that our main results hold: credit limits positively impact durations
(Column (1)), earnings replacement rates (Column(2)), and firm productivity (Column (4)).
Table 12: Over Identification Tests with Saiz and Gross & Souleles instrument. (Source:
2002-2006 LEHD/TransUnion)
(1)
(2)
—————–Over ID Tests with
Duration
Replacement
Rate
Dep Var
Unused Revolving Credit to Income Ratio
0.656***
(0.112)
Demographic, Industry, MSA, & Lagged Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
R2 (1st Stage for IVs)
0.137
Angirst Pischke FStat Pval
0
Pval Weak Id Null Weak
6.41e-05
Jtest Pval Null Valid
0.134
Round N
32000
(3)
(4)
Saiz-IV and GS-IV—————–
Large
Firm Productive Firm
Dummy
Dummy
0.0970***
(0.0157)
Y
0.0860
(0.0534)
Y
0.0987*
(0.0521)
Y
Y
0.120
0
4.46e-05
0.290
21000
Y
0.120
0
4.46e-05
0.0622
21000
Y
0.120
0
4.46e-05
0.230
21000
Notes. Clustered std. errors at MSA level in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Same control
definitions as Table 2.
B.2
Alternate Measures of Personal Financial Constraints: Total Credit, Revolving Credit Including HELOCs, and Credit
Scores
In Table 13, we use alternate endogenous regressors: (i) unused revolving credit to income,
including HELOCs (ii) total unused credit, including all types of secured (including HELOCs
and mortgage debt) and unsecured debt (we define ‘total unused credit to income’ as the
total credit limit less the amount currently borrowed over annual earnings, where the ratio
is measured 1 year prior to layoff)42 , and (iii) credit scores (this corresponds to TransUnions
bankruptcy model, and ranges from 0 to 1000, with higher scores indicating less credit risk).
Columns (1) and (2) of Table 13 illustrate that revolving used credit, inclusive of HELOCs,
has a similar effect on duration and replacement rates, respectively, as the baseline definition
42
The Total Credit Limit is formally the TransUnion variable “Total High Credit/Credit Limit” which
is sum of actual credit limits across all types of debt, or if the credit limit is not stated, it is the highest
observed prior balance. This measure of credit includes secured credit lines like home equity lines of credit
and installment credit, as well as auto loans, and other personal finance loans.
54
in the text (which excludes HELOCs). Likewise, Column (3) and (4) of Table 13 illustrate
that total unused credit has a similar effect on duration and replacement rates. Columns
(5) and (6) of Table 13 are more difficult to interpret, since the units are in terms of the TU
bankruptcy model, but in general, if an individual has a higher score prior to layoff, they
longer they take to find a job, and they find higher replacement rates, conditional on finding
a job.
Table 13: Alternate Endogenous Regressors, Saiz Instrument.
LEHD/TransUnion)
Saiz House Supply Elasticity
(1)
(2)
(3)
Dep Var
Duration
Replacement Duration
Rate
Unused Revolving Credit to Income 1.269**
0.162**
Ratio (Incl. HELOCS)
(0.525)
(0.0755)
Total Unused Credit to Income Ra1.508**
tio
(0.589)
Credit Score
(4)
Replacement
Rate
(Source:
2002-2007
(5)
Duration
(6)
Replacement
Rate
0.00238**
(0.000957)
Y
1.087***
(0.0665)
Y
0.134**
(0.0616)
Demographic, Industry, MSA, & Y
Y
Y
Y
Lagged Earnings Controls
R2 (1st Stage for IVs)
0.137
0.0896
0.104
0.117
0.110
0.113
Angirst Pischke FStat Pval
2.13e-05
1.39e-05
7.32e-05
4.48e-06
2.32e-05
2.78e-06
Round N
32000
21000
32000
21000
32000
21000
Notes. Clustered Std. errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Credit score is the lagged TU bankruptcy score.
Unused Revolving Credit to Income measured 1 year prior to layoff. Demographic controls include quadratic in age & tenure,
race, sex and education dummies as well as year & auto loan dummies. Industry controls include 1-digit SIC dummies and
lagged size, age, and wage per worker of prior firm. MSA controls include real per capita GDP and the MSA unemployment
rate. Lagged earnings controls include both lagged real annual earnings, and cumulative lagged real annual earnings to proxy
for assets. Equity proxy is highest observed mortgage balance less current mortgage balance. HELOC limits include combined
home equity limits.
55
B.3
Correlation of Unemployment Durations and Credit Limits
in the SCF, Controlling for Assets
In the SCF between 1998 and 2007 (which includes the 1998, 2001, 2004, and 2007 surveys), we can compute the raw correlation between unused credit limits and unemployment
durations, controlling for a host of assets, including home values. Figure 12 plots the raw
correlation between unemployment duration and credit limits in the SCF, and it reveals a
similar pattern to the LEHD/TransUnion dataset. Table 14 provides a more formal analysis,
including controls for the entire portfolio of a households assets. Table 14 demonstrates a
strong correlation between unused credit card limits and unemployment durations, subject
to attenuation bias. The ‘Unused Unsecured Limit to Income’ refers to unused credit card
limits (as of the survey date) over annual gross family income (over the prior year). Unemployment duration measures weeks spent unemployed over the past 12 months prior to the
survey. It is measured in weeks, and does not distinguish individual unemployment spells.
Column 1 of Table 14 shows that simple regressions of unemployment duration on unused credit card limits reveal a strong positive correlation, even after controlling for income
and liquid assets. Columns 2 and 3 impose age restrictions and add basic demographic controls, but the positive and significant relationship persists. Column 4 adds in all available
categories of illiquid assets, and finally Column 5 restricts the dataset to mortgagors (as
is the case in the LEHD/TransUnion sample considered in the text). The strong positive
and significant relationship between unused credit limits and unemployment durations persists. An unused credit limit worth 10% of prior annual family income is associated with 1
week longer unemployment spells, very similar to the IV estimate in the LEHD/TransUnion
sample considered in the text.
56
Figure 12: Survey of Consumer Finances: Correlation of Unemployment Durations (in
Weeks) on Unused Credit (Source: 1998-2007 SCF)
57
Table 14: Survey of Consumer Finances: OLS Regressions of Unemployment Durations
(in Weeks) on Unused Credit, Controlling for Assets (Source: 1998-2007 SCF)
(1)
(2)
(3)
(4)
(5)
(6)
—————— Dep. Var. is SCF Unemployment Durations in Weeks ——————
Unused Unsecured Limit to Income
12.334***
(5.85)
Year Dummies
N
Demographics and Income
N
Liquid Assets to Inc (Checking/Savings N
plus Stocks and Bonds)
Illiquid Assets to Inc (Homes, Vehicles, N
etc.)
Mortgagors Only
N
10.430***
(4.87)
8.733***
(4.02)
9.338***
(4.31)
8.155***
(3.75)
7.854***
(2.66)
Y
Y
N
Y
Y
Y
Y
Y
N
Y
Y
Y
Y
Y
Y
N
N
Y
Y
Y
N
N
N
N
Y
Observations
764
764
764
759
759
421
R-squared
0.052
0.130
0.144
0.137
0.148
0.157
Notes: SCF 24 to 65yo Heads of Household with Positive Unemployment Spell over Prior 12 months and Positive
Limit. Restrict to Mortgagors in Col 6. Demographics include quadratic in age, dummies for education, and dummies
for race and Income refers to gross annual family income. Liquid Assets include cash, checking, money market funds,
CDS, corporate bonds, government saving bonds, stocks, and mutual funds less credit card debt. Unused Credit Limit to
Income refers to total credit card limits less credit card balances. Illiquid Assets includes Homes, Vehicles, Retirement,
Annuities, Life Insurance at self-reported market values.
58
B.4
House Prices and the Relationship Between Credit, NonEmployment Durations, and Replacement Rates
Table 15 illustrates the main regressions estimated with OLS, with and without housing
price controls. The house price control we include in the regression is the OFHEO AllTransaction House Price Index for MSAs.43 We control for house prices at the time of
the layoff, and we find that the correlation between the unused credit limit ratio and nonemployment durations changes very little. Likewise, the replacement rate 1 year after layoff
is hardly impacted by the inclusion of house prices as controls. As discussed in the text, the
magnitude of the coefficients are significantly smaller than the IV estimates. This likely due
to the fact that high-type/high-earner households have higher amounts of credit on average,
but, orthogonal to credit access, take longer to find jobs (the PSID from 2005-2013 confirms
a strong significant negative relationship between duration and income). By isolating an
exogenous component of income, we mitigate this offsetting force due to largely unobserved
heterogeneity in worker types.
The fact that house prices increase does not necessarily imply households are wealthier,
nor that they should consume more, and this topic is actively debated in the literature
(see both sides of the literature in Calomiris et al. [2009], among others). As most theoretic
studies show, a household may attempt to sell the house, but they must buy a new one or rent
thereafter, mitigating housing wealth effects. There is also mixed evidence regarding housing
lock (see both sides of the literature discussed in Karahan and Rhee [2011]), suggesting that
housing wealth may not matter for selling decisions as well. We find very little evidence of
interstate movers or intrastate movers in our sample around job loss. If we drop movers, our
IV regression results remain unchanged in terms of sign, significance, and magnitude.
43
This is publicly avaiable from the OFHEO website. The index is normalized to 100 in 1995.
59
Table 15: Baseline OLS regressions with Direct Controls for OFHEO House Prices (Source:
LEHD/TransUnion 2002-2007)
(1)
Duration
(2)
Duration
(3)
Duration
(4)
(5)
(6)
Replacement Replacement Replacement
Rate
Rate
Rate
Unused Revolving Credit to Income 0.258***
Ratio
(0.0257)
0.248***
0.242***
0.0335***
0.0327***
0.0324***
(0.0258)
(0.0259)
(0.00344)
(0.00344)
(0.00345)
Demographic, Industry, MSA, & Y
Y
Y
Y
Y
Y
Lagged Earnings Controls
OFHEO HPI Index
N
Y
Y
N
Y
Y
HELOC Limits and Equity Proxy
N
N
Y
N
N
Y
R2
0.047
0.048
0.049
0.082
0.083
0.083
Round N
32000
32000
32000
21000
21000
21000
Notes. Clustered standard errors at MSA level in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Revolving Unused Credit
to Income measured 1 year prior to layoff. Demographic controls include quadratic in age & tenure, race, sex and education
dummies as well as year & auto loan dummies. Industry controls include 1-digit SIC dummies. MSA controls include real
per capita GDP and the MSA unemployment rate. Lagged earnings controls include both lagged real annual earnings, and
cumulative lagged real annual earnings to proxy for assets.
60
C
Replacement Earnings 2 Years After Layoff
Consider the set of households who find a job 1 year after layoff. In the main text, Table 5
illustrates the impact of credit on the wages of job finders 1 year after layoff. To assess the
impact of consumer credit access on longer term wage outcomes, Table 16 analyzes wages 2
years after layoff for this same sample. Within this sample, the OLS estimates in Column (1)
imply that replacement earnings are .3% higher, 2 years after layoff, for households who can
replace 10% more of their lost income with unused credit. The Saiz instrument in Column (2)
and the bankruptcy flag removal in Column (3) yield insignificant results, whereas the Gross
and Souleles estimates in Column (3) imply that within this sample, replacement earnings
are .9% higher, 2 years after layoff, for households who can replace 10% more of their lost
income with unused credit. While the results are mixed, the earnings gains in Table 5 persist
2 years after layoff for at least 2 of the specifications considered in the main text.
Table 16: Dependent Variable is Replacement Rate, Measured 2 Years After Layoff Relative
to 1 Year Before Layoff. Sample Restricted to Job Finders 1 Year After Layoff. (Source:
2002-2006 LEHD/TransUnion)
(1)
(2)
(3)
(4)
——— DV is Replacement Rate at t+2, Among Employed Sample ———
OLS
IV-Saiz
IV-GS
OLS-Flag Drop
Unused Revolving Credit to Income Ratio
0.0383***
0.180
0.0916***
(0.00441)
(0.122)
(0.0171)
Flag Drop (d)
-0.00598
(0.0136)
Demographic, Industry, MSA, & Lagged Y
Y
Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
Y
Y
R2 (1st Stage for IVs)
0.078
0.0872
0.117
0.120
Angirst Pischke FStat Pval
2.48e-05
0
Round N
21000
21000
21000
12000
Notes. Same as Table 2.
61
C.1
Self-Employment and the Earnings Gap Method
Table 17 redoes the main analysis in two different ways. Column (1) is a regression of
duration on unused credit where the self-employed with more than 5k in annual Schedule C
earnings are counted as employed. Column (2) infers the length of unemployment duration
using the earnings gap method. Using quarterly earnings prior to layoff as the base (Eq−1 ),
then those who find a job within the first quarter of layoff will have spent 1 − Eq /Eq−1
fraction of the quarter unemployed. Table 17 illustrates that the main results are robust to
these alternate definitions.
Table 17: Column (1) is duration of non-employment, counting the self-employed who earn
more than 5k in a year as employed, and Column (2) is duration of non-employment with partial duration values inferred using the earnings gap method. (Source: LEHD / TransUnion)
(1)
Duration (Self-Employment)
(2)
Duration (Earnings Gap Method)
Unused Revolving Credit to Income Ratio
1.334**
1.532**
(0.646)
(0.623)
Demographic, Industry, MSA, & Lagged Y
Y
Earnings Controls
HELOC Limits and Equity Proxy
Y
Y
R2 (1st Stage for IVs)
0.101
0.101
Angirst Pischke FStat Pval
0.000138
0.000138
Round N
32000
32000
Notes. Robust standard errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1. Revolving Unused Credit to Income
measured 1 year prior to layoff. Demographic controls include quadratic in age & tenure, race, sex and education
dummies as well as year & auto loan dummies. Industry controls include 1-digit SIC dummies. MSA controls include
real per capita GDP and the MSA unemployment rate. Lagged earnings controls include lagged real annual earnings.
C.2
Non-Linearities of Point Estimates
While not shown here, we have constructed several sets of results which stratify our samples
by tercile of income and tercile of initial debt levels. The lowest debt terciles exhibit the
greatest sensitivity to credit, consistent with our finding that previously bankrupt households
(who have very little debt and are credit constrained) have the greatest duration elasticities.
In terms of income, the middle tercile of the income distribution exhibits the strongest
duration and replacement rate elasticities. These results can be disclosed upon request.
62
D
Employed Value Functions
For employed households, value functions are denoted with a W , and at the end of every
period, employed households face layoff risk δ. If they are laid off, since the period is 1
quarter, we must allow the workers to search immediately for a new job.44
h
u(c,
0)
+
βE
(1 − δ)Wt+1 (b0 , h0 , k; Ω0 )
WtG (b, h, k; Ω) = max
b0 ≥b
+δ max p(θt+1 (h0 , k̃; Ω0 ))Wt+1 (b0 , h0 , k̃; Ω0 )
k̃
i
0
0
0
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (b , h , k; Ω ) , t ≤ T
WTG+1 (b, h, k; Ω) = 0
Such that the aggregate laws of motion are given by equation (4), human capital evolves
according to the law of motion below,
h0 = H(h, W )
and the budget constraint holds,
c + qW,t (b0 , h, k; Ω)b0 ≤ αf (y, h, k) + b
The value functions for employed borrowers who default as well as the discrete default
44
This allows the model to match labor flows in the data.
63
decision are formulated in an identical fashion to that of the unemployed.
h
WtB (b, h, k; Ω) = u(c, 0)+λβE (1 − δ)Wt+1 (0, h0 , k; Ω0 )
+ δ max p(θt+1 (h0 , k̃; Ω0 ))Wt+1 (0, h0 , k̃; Ω0 )
k̃
i
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (0, h , k; Ω )
h
B
+ (1 − λ)βE (1 − δ)Wt+1
(0, h0 , k; Ω0 )
B
(0, h0 , k̃; Ω0 )
+ δ max p(θt+1 (h0 , k̃; Ω0 ))Wt+1
k̃
i
B
0
0
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (0, h , k; Ω ) , t ≤ T
0
0
WTB+1 (b, h, k; Ω) = 0
Such that the aggregate laws of motion are given by equation (4), human capital evolves
such that h0 = H(h, W ) and the budget constraint is given by,
c ≤ αf (y, h, k)
For employed households in good standing, at the start of every period, they must make the
following default decision,
n
o
Wt (b, h, k; Ω) = max WtG (b, h, k; Ω), WtB (b, h, k; Ω) − χ
Let DW,t (b, h, k; Ω) denote the employed household’s default decision.
E
Proofs
Restatement of Proposition 5.1: Assume that the utility function meets standard conditions (u0 > 0, u00 < 0, limc→0 u0 (c) = ∞, limc→∞ u0 (c) = 0, and u is invertible), the matching
function is invertible and constant returns to scale, and there is a bounded support (which can
be non-binding) for the choice set of debt b ∈ B ⊆ [b, b] and the capital of firms k ∈ K ⊆ [k, k],
then a Block Recursive Equilibrium exists.
64
Proof. The proof will follow backward induction. Let t = T , and consider an unemployed
household for the sake of brevity (an identical argument follows for employed households).
Since the household’s continuation value is zero from T + 1 onward, the household dynamic
programming problem trivially does not depend on the aggregate distribution µ across states
in the last period of life,
UTG (b, h, k; Ω) = u(z(k) + b, 1) + β · 0
= UTG (b, h, k; y, b)
WTG (b, h, k; Ω) = u(αf (y, h, k) + b, 1) + β · 0
= WTG (b, h, k; y, b)
In this last period of life, the saving and borrowing policy function b0e,T (b, h, k; y, b) is trivially
zero (for both employed e = W and unemployed agents e = U ). Likewise, for households
in bad standing in the last period of life, the value of unemployment (and nearly identical
conditions hold for the employed, and so are omitted) is given by,
UtB (b, h, k; y, b) = u(z(k), 1) + β · 0
Stepping back to the default decision, UT will also not depend on the aggregate distribution
µ,
n
o
G
B
UT (b, h, k; y, b) = max UT (b, h, k; y, b), UT (0, h, k; y, b) − χ
Let DU,T (b, h, k; y, b) denote the policy function of the household. Since there is a utility
penalty χ of defaulting, debt can be supported in equilibrium, and DU,T will not be trivially
zero.
Now stepping back to the labor search problem, the firm’s value function will be independent of µ as well,
JT (h, k; Ω) = (1 − α)f (y, h, k) + β · 0
= JT (h, k; y, b)
65
And the labor market tightness will also be independent of µ,
θT (h, k; Ω) = p−1
f
κ + (1 + rf )k
JT (h, k; y, b)
!
= θT (h, k; y, b)
The household at age T − 1 (note that the primes below simply note that age T − 1 risk
over y and b has already been realized and human capital has already evolved to h0 ) must
therefore make the following labor market search choice over k, the capital of firms,
max p(θT (h0 , k; y 0 , b0 ))WT (b0 , h0 , k; y 0 , b0 ) + 1 − p(θT (h0 , k; y 0 , b0 )) UT (b0 , h0 , k; y 0 , b0 )
k∈K
(12)
So long as k lies in a bounded interval, the extreme value theorem guarantees at least one
solution to this problem. As we will see below, for certain classes of production functions,
only one solution exists. For the current exposition, assume the production function lies
within this class, and a unique solution exists.
0
(h0 , k; y 0 , b0 ),
Given the household policy functions for labor search kT0 −1 (h0 , k; y 0 , b0 ) and default De,T
the bond price qU,T (b0 , h, k; Ω) is given by,
qU,T −1 (b0 , h, k; Ω) =
h
i
E 1 − De0 ,T (b0 , h0 , k 0 ; y 0 , b0 )
1 + rf
= qU,T −1 (b0 , h, k; y, b)
Clearly the bond price does not depend on the aggregate distribution µ.
Stepping back from t = T − 1, . . . , 1, and repeating the above procedure completes the
proof.
Restatement of Corollary 5.2: Under the hypotheses of Proposition 5.1, so long as
χ > 0, and B contains a neighborhood of debt around 0, a Block Recursive Equilibrium with
credit exists.
Proof. Because of the inada conditions, for every positive χ ∈ R+ , there exists a sufficiently
66
small debt in an -neighborhood around zero, b ∈ N (0), such that the household strictly
prefers repayment in the last period of life. The households repayment choice is given by,
o
n
max UTG (b, h, k; y, b), UTB (0, h, k; y, b) − χ
This holds with equality at the cutoff debt b∗ ,
UTG (b∗ , h, k; y, b) = UTB (0, h, k; y, b) − χ
Substituting,
u(z(k) + b∗ , 1) = u(z(k), 1) − χ
The minimum supportable debt is given by,
b∗ = u−1 (u(z(k), 1) − χ, 1) − z(k) < 0
Restatement of Lemma (5.3): In addition to the assumptions in Proposition 5.1,
let the production function be Cobb-Douglas, i.e. f (y, h, k) = yh1−a k a (0 < a < 1), let the
1
1
matching function be given by M (u, v) = u 2 v 2 , let χ → ∞ (no default for households), the
value of leisure is zero, and assume there is no uncertainty over human capital h, aggregate
productivity y, or the borrowing limit b. Then if the utility function is negative, increasing,
1−σ
and concave (e.g. c 1−σ−1 for σ > 1 or u(c) = −e−c ), the household labor search problem
(equation (12)) admits a unique solution.
Proof. The non-stochastic firm problem can be solved by hand, and under the hypotheses
of the present lemma, it is directly proportional to capital,
Jt (h, k) =
(1 − α)f (y, h, k) (β(1 − δ))T −t+1 (1 − α)f (y, h, k)
−
∝ ka
1 − β(1 − δ)
1 − β(1 − δ)
1
1
Under the assumption M (u, v) = u 2 v 2 , the equilibrium market tightness θt (h, k) can be
solved by hand.
"
1
κ = θt (h, k; Ω)− 2 Jt (h, k; Ω) − (1 + rf ) ·
67
#
k
1
θt (h, k; Ω)− 2
Solving for θt yields,
κ + (1 + rf )k
Jt (h, k; Ω)
!−2
= θt (h, k; Ω)
The household job finding rate is therefore given by,
Jt (h, k; Ω)
κ + (1 + rf )k
!
= p(θt (h, k; Ω))
For κ and rf sufficiently small,
p(θt (h, k; Ω)) ∝ k a−1
The constant worker share α in combination with the non-negative and increasing production
function implies that the wage a worker receives is concave and increasing in k. Note that
the composition of two non-decreasing concave functions in k preserves concavity in k, i.e.
ũ(k) = u(w(h, k) + µ) is concave in k for arbitrary µ. Let u be the outside option of the
household if they remain unemployed. Since the probability of finding a job is directly
proportional to k a−1 , the household chooses k to maximize
k a−1 ũ(k) + (1 − k a−1 )u
Since −k a−1 is concave, we ignore the second term (the idea will be to show the first term
is concave, and then use the fact that the sum of two concave functions is concave). The
condition for the first term to be concave is given by,
(a − 1)(a − 2)k a−3 ũ(k) + 2(a − 1)k a−2 ũ0 (k) + k a−1 ũ00 (k) < 0
|
{z
} |
{z
} | {z }
(−)
(−)
(−)
Under the hypotheses that u < 0, u0 > 0, u00 < 0 (note, these properties transfer to ũ), and
0 < a < 1, the labor search problem of the household is strictly concave and one solution
exists for k.
68
F
Business Cycle Moments
Table 18 displays the business cycle moments for the main model in the text versus the data.
The table makes the shortcomings of the model quite clear: the model is unresponsive to
productivity shocks (Shimer [2005] and more recently Chodorow-Reich and Karabarbounis
[2013]). Why does the Hagedorn and Manovskii [2008] calibration not work in this context?
They noticed that the flow utility from non-employment must be large enough to make
workers nearly indifferent between working and not working: workers then become sensitive
to small movement in productivity and wages. It is impossible to make every type of worker
indifferent between working and not working with significant heterogeneity and a constant
unemployment benefit or flow utility of leisure. The only paper to our knowledge to address
this issue is Lise and Robin [2013] who make the flow utility of non-employment a function
of the workers type, the workers type squared, the aggregate state, and interactions between
the workers type and the aggregate state. One alternate approach used by Christiano et al.
[2013] to generate cyclical responses in the economy could be to squeeze firm surplus so that
vacancy posting becomes very sensitive to small movements in productivity. However, since
firms are heterogeneous as well, only the lowest type firm will be sensitive to productivity
movements.
69
Table 18: Business Cycle Moments for Model During Main Simulation (1974 to 2012) vs.
Data
Model
x
SD(x)/SD(y)
Autocorr(x)
Corr(·,x)
u
Data
x
SD(x)/SD(y)
Autocorrelation
Corr(·,x)
u1
u1
v
θ
y
k̃
0.97
0.79
UE
Rate
1.66
0.40
Default
Rate*
0.00
0.13
2.69
0.73
1.66
0.39
1.91
0.78
1.00
0.82
1.00
-0.19
-0.74
-0.72
-0.73
-0.87
-0.14
u1
v
θ
y
k̃
10.10
0.94
19.10
0.94
1.00
0.88
-
UE
Rate
5.90
0.91
Default
Rate*
6.07
0.92
9.50
0.94
1.00
-0.89
-0.97
-0.41
-
-0.95
0.55
Notes: HP filtered with smoothing parameter 105 to be consistent with Shimer [2005]. Data
are from Shimer [2005], except (*) the default rate which is taken from Equifax (1999-2012).
As in the data, u1 is calculated as the fraction of unemployed households at the end of a
v
quarter. θ = u1 +u
includes the measure of households that immediately found jobs (u2 ),
2
hence the low volatility as that mass is quite large and very stable.
70
G
Model Robustness: Capital Investment and Liquidation
G.1
Model with Firm Investment
Now assume that Firms can invest in capital, depending on the worker’s type. The problem
of an unemployed household is unchanged. The value functions for employed borrowers who
default as well as the discrete default decision are formulated in an identical fashion to that
of the unemployed.
Timing assumption: New capital is not operable immediately.
As a result, the Bellman equation for a household in bad standing is given below (good
standing is extremely similar):
h
WtB (b, h, k; Ω) = u(c, 0)+λβE (1 − δ)Wt+1 (0, h0 , k’; Ω0 )
+ δ max p(θt+1 (h0 , k̃; Ω0 ))Wt+1 (0, h0 , k̃; Ω0 )
k̃
i
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (0, h , k; Ω )
h
B
+ (1 − λ)βE (1 − δ)Wt+1
(0, h0 , k’; Ω0 )
B
+ δ max p(θt+1 (h0 , k̃; Ω0 ))Wt+1
(0, h0 , k̃; Ω0 )
k̃
i
B
0
0
0
0
+ 1 − p(θt+1 (h , k̃; Ω )) Ut+1 (0, h , k; Ω ) , t ≤ T
0
0
WTB+1 (b, h, k; Ω) = 0
Such that the aggregate laws of motion are given by equation (4), human capital evolves
such that h0 = H(h, W ) and the budget constraint is given by,
c ≤ αf (y, h, k)
71
And, additionally
0
k 0 = kt∗ (h, k; Ω)
0
This final condition k 0 = kt∗ (h, k; Ω) means that households have rational expectations
over what the entrepreneurs’s optimal investment decision is.
G.2
Lenders
Lenders’ bond prices are update to reflect changes in capital, since it may affect the wage of
the worker and hence their repayment probability.
G.3
Entrepreneurs
We now allow entrepreneurs to invest in capital subject to an adjustment cost Γ(k 0 − k).
Therefore the value function for the firm is given by,
0
0 0
0
Jt (h, k; Ω) = max
(1
−
α)f
(y,
h,
k)
−
i
−
Γ(k
−
k)
−
f
+
βE
(1
−
δ)J
(h
,
k
;
Ω
)
c
t+1
0
k
Subject to a unit investment cost (i.e. the MRT of output and capital is 1, excluding the
adjustment cost),
i = k0 − k
JT +1 (h, k; Ω) = 0
In the results below, we choose a quadratic adjustment cost Γ(x) = x2 . We see that
the presence of firm investment does not significantly alter the main set of results. Figure
13 illustrates employment with the quadratic adjustment cost, Figure 14 illustrates sorting
(the correlation between human capital and capital), Figure 15 illustrates firm capital, and
Figure 16 illustrates labor productivity. Figure 15 shows that firm capital recovers much
faster as firms invest in more capital per worker as productivity recovers and human capital
grows.
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Figure 13: Allowing for Capital Investment: Employment
Figure 14: Allowing for Capital Investment: Corr. B/w Human Capital (h) and
Firm Capital (k)
Figure 15: Allowing for Capital Investment: Agg. Firm Capital, 2007-2009 Recession
Figure 16: Allowing for Capital Investment: Labor Productivity, 2007-2009 Recession
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G.4
Liquidation
We also allow for the baseline model to have a liquidation value of capital, χf . The continuation value of the firm becomes,
Jt (h, k; Ω) = (1 − α)f (y, h, k) − fc + βE (1 − δ)Jt+1 (h0 , k; Ω0 ) + δχf k
In the results below, we choose χf = .25 which is relatively low, but it allows us to preserve
the calibration, approximately. For larger values of χf , the same aggregate patterns emerge,
except we must significantly expand the capital grid to a point that it become computationally infeasible. Figures 17 and 18 illustrate the model’s main results with liquidation values.
Employment rises while productivity falls in both cases, which is the same pattern that
emerged when tighter debt limits were imposed in an economy with no liquidation value.
Figure 18: Liquidation Value Experiment: Labor Productivity, 2007-2009 Recession
Figure 17: Liquidation Value Experiment: Employment, 2007-2009 Recession
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H
Solution Algorithm
We solve the model using value function iteration on a discrete grid. Capital lies between
[0.25,1] with 40 evenly spaced grid points including the ends of the grid. Bonds lies on the
grid [-.5,1.5] with 81 evenly spaced grid points. With the discount factor given in the paper,
.04% of all simulated agent-time observations ever hit the top of the asset grid. The human
capital grid is 6 evenly spaced grid points including the end of the grid over [.5,1]. The
aggregate shock is discretized with 4 states using Rouwenhorst’s method. The bond limit is
discretized with 2 possible values.
Starting at t = T and working backwards, the solution method is given below:
i. Recover Jt (h, k; Ω) using value function iteration.
!
ii. Recover θt (h, k; Ω), the market tightness, by free entry, θt (h, k; Ω) = p−1
f
κ+(1+rf )k
Jt (h,k;Ω)
iii. Solve the household default decision to recover De,t (b, h, k; Ω).
iv. Solve the household maximization problem over the grid of k’s to recover kt (b, h, k; Ω)
using the market tightness and the implied job finding rates in step ii.
v. Use realized search behavior and default outcomes to recover the bond price qe,t (b, h, k; Ω)
(in the last period of life, this is simply zero).
vi. Solve the household maximization problem over the grid of b’s to recover b0e,t (b, h, k; Ω),
taking the bond price from step v as given.
vii. Repeat i to vi until t=1.
viii. Fix the aggregate shock path. Use policy functions from household problem to simulate
30,000 households for 270 periods, 10 times, burning the first 100 periods. Report
average over the 10 simulations.
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